{"id":4093,"date":"2025-06-28T23:32:17","date_gmt":"2025-06-29T04:32:17","guid":{"rendered":"https:\/\/www.ezcalculatoronline.com\/?page_id=4093"},"modified":"2026-05-12T23:22:49","modified_gmt":"2026-05-13T04:22:49","slug":"bilinear-interpolation-calculator","status":"publish","type":"page","link":"https:\/\/www.ezcalculatoronline.com\/math\/bilinear-interpolation-calculator\/","title":{"rendered":"Bilinear Interpolation Calculator"},"content":{"rendered":"<div id=\"tdi_1\" class=\"tdc-zone\"><div class=\"tdc_zone tdi_2  wpb_row td-pb-row\"  >\n<style scoped>\n\n\/* custom css *\/\n.tdi_2{\r\n                    min-height: 0;\r\n                }\n<\/style><div id=\"tdi_3\" class=\"tdc-row\"><div class=\"vc_row tdi_4  wpb_row td-pb-row\" >\n<style scoped>\n\n\/* custom css *\/\n.tdi_4,\r\n                .tdi_4 .tdc-columns{\r\n                    min-height: 0;\r\n                }.tdi_4,\r\n\t\t\t\t.tdi_4 .tdc-columns{\r\n\t\t\t\t    display: block;\r\n\t\t\t\t}.tdi_4 .tdc-columns{\r\n\t\t\t\t    width: 100%;\r\n\t\t\t\t}\n<\/style><div class=\"vc_column tdi_6  wpb_column vc_column_container tdc-column td-pb-span8\">\n<style scoped>\n\n\/* custom css *\/\n.tdi_6{\r\n                    vertical-align: baseline;\r\n                }.tdi_6 > .wpb_wrapper,\r\n\t\t\t\t.tdi_6 > .wpb_wrapper > .tdc-elements{\r\n\t\t\t\t    display: block;\r\n\t\t\t\t}.tdi_6 > .wpb_wrapper > .tdc-elements{\r\n\t\t\t\t    width: 100%;\r\n\t\t\t\t}.tdi_6 > .wpb_wrapper > .vc_row_inner{\r\n\t\t\t\t    width: auto;\r\n\t\t\t\t}.tdi_6 > .wpb_wrapper{\r\n\t\t\t\t    width: auto;\r\n\t\t\t\t    height: auto;\r\n\t\t\t\t}\n<\/style><div class=\"wpb_wrapper\" ><div class=\"wpb_wrapper td_block_wrap vc_raw_html tdi_8 \"><div class=\"td-fix-index\"><!DOCTYPE html>\n<html lang=\"en\">\n<head>\n  <meta charset=\"UTF-8\" \/>\n  <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\" \/>\n  <title>Bilinear Interpolation Calculator<\/title>\n  <style>\n    body {\n      font-family: 'Inter', sans-serif;\n      background: #eef2f3;\n      flex-direction: column;\n      align-items: center;\n    }\n\n    .container {\n      background: #ffffff;\n      border-radius: 16px;\n      box-shadow: 0 10px 30px rgba(0, 0, 0, 0.1);\n      max-width: 700px;\n      width: 100%;\n      padding: 30px;\n    }\n\n    h1 {\n      text-align: center;\n      margin-bottom: 25px;\n      color: #333;\n    }\n\n    h2 {\n      font-size: 18px;\n      margin: 25px 0 10px;\n      color: #0056b3;\n    }\n\n    .grid {\n      display: grid;\n      grid-template-columns: repeat(auto-fit, minmax(140px, 1fr));\n      gap: 15px;\n    }\n\n    label {\n      font-weight: 600;\n      display: block;\n      margin-bottom: 6px;\n      color: #333;\n    }\n\n    input[type=\"number\"] {\n      width: 100%;\n      padding: 8px;\n      font-size: 15px;\n      border: 1px solid #ccc;\n      border-radius: 8px;\n    }\n\n    button {\n      width: 100%;\n      padding: 12px;\n      font-size: 16px;\n      background-color: #007bff;\n      color: white;\n      border: none;\n      border-radius: 8px;\n      cursor: pointer;\n      margin-top: 25px;\n    }\n\n    #result {\n      margin-top: 20px;\n      background-color: #f1f8ff;\n      padding: 16px;\n      border-radius: 10px;\n      color: #0a2c5e;\n      font-weight: 600;\n      text-align: center;\n      display: none;\n    }\n\n    canvas {\n      margin: auto;\n      margin-top: 30px;\n      background: #fff;\n      border: 1px solid #ccc;\n      border-radius: 8px;\n      display: none;\n    }\n  <\/style>\n<\/head>\n<body>\n  <div class=\"container\">\n    <h1>Bilinear Interpolation Calculator<\/h1>\n    <form id=\"interpolationForm\">\n      <h2>\ud83e\udded Coordinates of Interpolation Rectangle<\/h2>\n      <div class=\"grid\">\n        <div><label>X1<\/label><input type=\"number\" id=\"x1\" value=\"0\" required><\/div>\n        <div><label>X2<\/label><input type=\"number\" id=\"x2\" value=\"10\" required><\/div>\n        <div><label>Y1<\/label><input type=\"number\" id=\"y1\" value=\"0\" required><\/div>\n        <div><label>Y2<\/label><input type=\"number\" id=\"y2\" value=\"10\" required><\/div>\n      <\/div>\n\n      <h2>\ud83d\udcca Values at 4 Corners<\/h2>\n      <div class=\"grid\">\n        <div><label>Q11 (x1, y1)<\/label><input type=\"number\" id=\"q11\" value=\"5\" required><\/div>\n        <div><label>Q12 (x1, y2)<\/label><input type=\"number\" id=\"q12\" value=\"15\" required><\/div>\n        <div><label>Q21 (x2, y1)<\/label><input type=\"number\" id=\"q21\" value=\"20\" required><\/div>\n        <div><label>Q22 (x2, y2)<\/label><input type=\"number\" id=\"q22\" value=\"30\" required><\/div>\n      <\/div>\n\n      <h2>\ud83d\udccc Interpolate At Point<\/h2>\n      <div class=\"grid\">\n        <div><label>X<\/label><input type=\"number\" id=\"x\" value=\"5\" required><\/div>\n        <div><label>Y<\/label><input type=\"number\" id=\"y\" value=\"5\" required><\/div>\n      <\/div>\n\n      <button type=\"button\" onclick=\"calculate()\">Calculate<\/button>\n    <\/form>\n\n    <div id=\"result\"><\/div>\n    <canvas id=\"graph\" width=\"400\" height=\"400\"><\/canvas>\n  <\/div>\n\n  <script>\n    function calculate() {\n      const x1 = parseFloat(document.getElementById(\"x1\").value);\n      const x2 = parseFloat(document.getElementById(\"x2\").value);\n      const y1 = parseFloat(document.getElementById(\"y1\").value);\n      const y2 = parseFloat(document.getElementById(\"y2\").value);\n      const q11 = parseFloat(document.getElementById(\"q11\").value);\n      const q12 = parseFloat(document.getElementById(\"q12\").value);\n      const q21 = parseFloat(document.getElementById(\"q21\").value);\n      const q22 = parseFloat(document.getElementById(\"q22\").value);\n      const x = parseFloat(document.getElementById(\"x\").value);\n      const y = parseFloat(document.getElementById(\"y\").value);\n\n      if (x1 === x2 || y1 === y2) {\n        document.getElementById(\"result\").innerText = \"Error: x1 \u2260 x2 and y1 \u2260 y2 are required.\";\n        document.getElementById(\"result\").style.display = 'block';\n        document.getElementById(\"graph\").style.display = 'none';\n        return;\n      }\n\n      const fxy1 = ((x2 - x) \/ (x2 - x1)) * q11 + ((x - x1) \/ (x2 - x1)) * q21;\n      const fxy2 = ((x2 - x) \/ (x2 - x1)) * q12 + ((x - x1) \/ (x2 - x1)) * q22;\n      const fxy = ((y2 - y) \/ (y2 - y1)) * fxy1 + ((y - y1) \/ (y2 - y1)) * fxy2;\n\n      document.getElementById(\"result\").innerHTML = `\ud83d\udd0d Interpolated value at (x = <strong>${x}<\/strong>, y = <strong>${y}<\/strong>) is:<br><strong style=\"font-size: 20px;\">${fxy.toFixed(4)}<\/strong>`;\n      document.getElementById(\"result\").style.display = 'block';\n      document.getElementById(\"graph\").style.display = 'block';\n\n      drawCanvas(x1, x2, y1, y2, x, y);\n    }\n\n    function drawCanvas(x1, x2, y1, y2, x, y) {\n      const canvas = document.getElementById(\"graph\");\n      const ctx = canvas.getContext(\"2d\");\n      ctx.clearRect(0, 0, canvas.width, canvas.height);\n\n      const margin = 40;\n      const scaleX = (canvas.width - 2 * margin) \/ (x2 - x1);\n      const scaleY = (canvas.height - 2 * margin) \/ (y2 - y1);\n\n      function mapX(val) {\n        return margin + (val - x1) * scaleX;\n      }\n\n      function mapY(val) {\n        return canvas.height - (margin + (val - y1) * scaleY);\n      }\n\n      ctx.strokeStyle = \"#007bff\";\n      ctx.lineWidth = 2;\n      ctx.strokeRect(mapX(x1), mapY(y2), (x2 - x1) * scaleX, (y2 - y1) * scaleY);\n\n      const points = [\n        { x: x1, y: y1, label: \"Q11\" },\n        { x: x2, y: y1, label: \"Q21\" },\n        { x: x1, y: y2, label: \"Q12\" },\n        { x: x2, y: y2, label: \"Q22\" }\n      ];\n\n      ctx.fillStyle = \"#007bff\";\n      ctx.font = \"14px Inter\";\n      for (const pt of points) {\n        ctx.beginPath();\n        ctx.arc(mapX(pt.x), mapY(pt.y), 5, 0, Math.PI * 2);\n        ctx.fill();\n        ctx.fillText(pt.label, mapX(pt.x) + 8, mapY(pt.y) - 8);\n      }\n\n      ctx.fillStyle = \"red\";\n      ctx.beginPath();\n      ctx.arc(mapX(x), mapY(y), 6, 0, Math.PI * 2);\n      ctx.fill();\n      ctx.fillText(\"P(x,y)\", mapX(x) + 10, mapY(y) - 8);\n    }\n  <\/script>\n<\/body>\n<\/html>\n<\/div><\/div><div class=\"wpb_wrapper wpb_text_column td_block_wrap td_block_wrap vc_column_text tdi_9  tagdiv-type td-pb-border-top td_block_template_1\"  data-td-block-uid=\"tdi_9\" ><div class=\"td-block-title-wrap\"><\/div><div class=\"td-fix-index\"><h1 data-start=\"157\" data-end=\"215\">\ud83d\udcd8 <strong data-start=\"164\" data-end=\"215\">Introduction: Bilinear Interpolation Calculator<\/strong><\/h1>\n<h2 data-start=\"217\" data-end=\"511\"><strong data-start=\"217\" data-end=\"252\">What is Bilinear Interpolation?<\/strong><\/h2>\n<p data-start=\"217\" data-end=\"511\">Bilinear interpolation is a mathematical method used to estimate values at a point within a rectangular area, based on the values at the four surrounding corner points. It is widely used in image processing, geographical mapping, and numerical simulations.<\/p>\n<h2 data-start=\"513\" data-end=\"723\"><strong data-start=\"513\" data-end=\"538\">About This Calculator<\/strong><\/h2>\n<p data-start=\"513\" data-end=\"723\">This Bilinear Interpolation Calculator helps you quickly compute the interpolated value at any point (x, y) inside a rectangle, based on the known values at the four corner points:<\/p>\n<ul data-start=\"724\" data-end=\"801\">\n<li data-start=\"724\" data-end=\"743\">\n<p data-start=\"726\" data-end=\"743\">Q11 at (x1, y1)<\/p>\n<\/li>\n<li data-start=\"744\" data-end=\"763\">\n<p data-start=\"746\" data-end=\"763\">Q21 at (x2, y1)<\/p>\n<\/li>\n<li data-start=\"764\" data-end=\"783\">\n<p data-start=\"766\" data-end=\"783\">Q12 at (x1, y2)<\/p>\n<\/li>\n<li data-start=\"784\" data-end=\"801\">\n<p data-start=\"786\" data-end=\"801\">Q22 at (x2, y2)<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"803\" data-end=\"1014\">Just input the coordinates and values, and the tool will instantly calculate the interpolated result. It also includes a visual 2D graph to help you understand how the interpolation is performed within the grid.<\/p>\n<h2 data-start=\"1016\" data-end=\"1029\"><strong data-start=\"1016\" data-end=\"1029\">Features:<\/strong><\/h2>\n<ul data-start=\"1030\" data-end=\"1254\">\n<li data-start=\"1030\" data-end=\"1064\">\n<p data-start=\"1032\" data-end=\"1064\">Clean and responsive interface<\/p>\n<\/li>\n<li data-start=\"1065\" data-end=\"1112\">\n<p data-start=\"1067\" data-end=\"1112\">Grouped input fields for easy understanding<\/p>\n<\/li>\n<li data-start=\"1113\" data-end=\"1152\">\n<p data-start=\"1115\" data-end=\"1152\">Automatic interpolation calculation<\/p>\n<\/li>\n<li data-start=\"1206\" data-end=\"1254\">\n<p data-start=\"1208\" data-end=\"1254\">Ideal for students, engineers, and researchers<\/p>\n<\/li>\n<\/ul>\n<h2 data-start=\"93\" data-end=\"151\"><strong data-start=\"99\" data-end=\"151\">How to Use the Bilinear Interpolation Calculator<\/strong><\/h2>\n<ol data-start=\"153\" data-end=\"1001\">\n<li data-start=\"153\" data-end=\"408\">\n<h3 data-start=\"156\" data-end=\"205\">\ud83e\udded <strong data-start=\"163\" data-end=\"205\">Enter the Coordinates of the Rectangle<\/strong><\/h3>\n<ul data-start=\"209\" data-end=\"408\">\n<li data-start=\"209\" data-end=\"408\">\n<p data-start=\"211\" data-end=\"272\">These define the corners of the grid (must form a rectangle):<\/p>\n<ul data-start=\"278\" data-end=\"408\">\n<li data-start=\"278\" data-end=\"321\">\n<p data-start=\"280\" data-end=\"321\"><strong data-start=\"280\" data-end=\"286\">X1<\/strong> and <strong data-start=\"291\" data-end=\"297\">X2<\/strong> (horizontal boundaries)<\/p>\n<\/li>\n<li data-start=\"327\" data-end=\"408\">\n<p data-start=\"329\" data-end=\"408\"><strong data-start=\"329\" data-end=\"335\">Y1<\/strong> and <strong data-start=\"340\" data-end=\"346\">Y2<\/strong> (vertical boundaries)<br data-start=\"368\" data-end=\"371\" \/>\ud83d\udc49 <em data-start=\"377\" data-end=\"408\">Make sure X1 \u2260 X2 and Y1 \u2260 Y2<\/em><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"410\" data-end=\"679\">\n<h3 data-start=\"413\" data-end=\"455\">\ud83d\udcca <strong data-start=\"420\" data-end=\"455\">Enter the Values at Each Corner<\/strong><\/h3>\n<ul data-start=\"459\" data-end=\"679\">\n<li data-start=\"459\" data-end=\"679\">\n<p data-start=\"461\" data-end=\"533\">These are the known values (often from measurements or image pixels) at:<\/p>\n<ul data-start=\"539\" data-end=\"679\">\n<li data-start=\"539\" data-end=\"570\">\n<p data-start=\"541\" data-end=\"570\"><strong data-start=\"541\" data-end=\"548\">Q11<\/strong> \u2192 Value at (X1, Y1)<\/p>\n<\/li>\n<li data-start=\"576\" data-end=\"607\">\n<p data-start=\"578\" data-end=\"607\"><strong data-start=\"578\" data-end=\"585\">Q21<\/strong> \u2192 Value at (X2, Y1)<\/p>\n<\/li>\n<li data-start=\"613\" data-end=\"644\">\n<p data-start=\"615\" data-end=\"644\"><strong data-start=\"615\" data-end=\"622\">Q12<\/strong> \u2192 Value at (X1, Y2)<\/p>\n<\/li>\n<li data-start=\"650\" data-end=\"679\">\n<p data-start=\"652\" data-end=\"679\"><strong data-start=\"652\" data-end=\"659\">Q22<\/strong> \u2192 Value at (X2, Y2)<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"681\" data-end=\"806\">\n<h3 data-start=\"684\" data-end=\"724\">\ud83d\udccc <strong data-start=\"691\" data-end=\"724\">Enter the Target Point (x, y)<\/strong><\/h3>\n<ul data-start=\"728\" data-end=\"806\">\n<li data-start=\"728\" data-end=\"806\">\n<p data-start=\"730\" data-end=\"806\">This is the point inside the rectangle where you want to estimate the value.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"808\" data-end=\"1001\">\n<h3 data-start=\"811\" data-end=\"839\">\u25b6\ufe0f <strong data-start=\"818\" data-end=\"839\">Click \u201cCalculate\u201d<\/strong><\/h3>\n<ul data-start=\"843\" data-end=\"1001\">\n<li data-start=\"843\" data-end=\"1001\">\n<p data-start=\"845\" data-end=\"909\">The tool will apply the bilinear interpolation formula and show:<\/p>\n<ul data-start=\"915\" data-end=\"1001\">\n<li data-start=\"915\" data-end=\"944\">\n<p data-start=\"917\" data-end=\"944\">The <strong data-start=\"921\" data-end=\"944\">interpolated result<\/strong><\/p>\n<\/li>\n<li data-start=\"950\" data-end=\"1001\">\n<p data-start=\"952\" data-end=\"1001\">A <strong data-start=\"954\" data-end=\"966\">2D graph<\/strong> with your point and corners marked<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<h2 data-start=\"1008\" data-end=\"1023\"><strong>\ud83d\udcc8 Example:<\/strong><\/h2>\n<p data-start=\"1024\" data-end=\"1123\">If you want to estimate a value inside a rectangle bounded by (0,0) and (10,10) with corner values:<\/p>\n<p data-start=\"1024\" data-end=\"1123\">Q11 = 5\u00a0 \u00a0 \u00a0Q21 = 20<br \/>\nQ12 = 15\u00a0 \u00a0Q22 = 30<\/p>\n<h2 data-start=\"0\" data-end=\"41\"><strong data-start=\"7\" data-end=\"41\">Bilinear Interpolation Formula<\/strong><\/h2>\n<p data-start=\"43\" data-end=\"182\">Bilinear interpolation estimates the value of a point <strong data-start=\"97\" data-end=\"107\">(x, y)<\/strong> inside a rectangle using the known values at the rectangle\u2019s four corners.<\/p>\n<h3 data-start=\"189\" data-end=\"206\">\u00a0<strong data-start=\"196\" data-end=\"206\">Given:<\/strong><\/h3>\n<ul data-start=\"208\" data-end=\"480\">\n<li data-start=\"208\" data-end=\"270\">\n<p data-start=\"210\" data-end=\"228\">Rectangle corners:<\/p>\n<ul data-start=\"231\" data-end=\"270\">\n<li data-start=\"231\" data-end=\"270\">\n<p data-start=\"233\" data-end=\"270\">\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(x_1, y_1)<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord\"><span class=\"mord mathnormal\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span> and<\/p>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>y<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(x_2, y_2)<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord\"><span class=\"mord mathnormal\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li data-start=\"271\" data-end=\"451\">\n<p data-start=\"273\" data-end=\"291\">Values at corners:<\/p>\n<ul data-start=\"294\" data-end=\"451\">\n<li data-start=\"294\" data-end=\"331\">\n<p data-start=\"296\" data-end=\"331\">\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>Q<\/mi><mn>11<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">Q_{11}<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">Q<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">11<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>: at<\/p>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(x_1, y_1)<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord\"><span class=\"mord mathnormal\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/li>\n<li data-start=\"334\" data-end=\"371\">\n<p data-start=\"336\" data-end=\"371\">\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>Q<\/mi><mn>21<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">Q_{21}<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">Q<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">21<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>: at<\/p>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(x_2, y_1)<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord\"><span class=\"mord mathnormal\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/li>\n<li data-start=\"374\" data-end=\"411\">\n<p data-start=\"376\" data-end=\"411\">\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>Q<\/mi><mn>12<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">Q_{12}<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">Q<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">12<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>: at<\/p>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>y<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(x_1, y_2)<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord\"><span class=\"mord mathnormal\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/li>\n<li data-start=\"414\" data-end=\"451\">\n<p data-start=\"416\" data-end=\"451\">\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>Q<\/mi><mn>22<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">Q_{22}<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">Q<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">22<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>: at<\/p>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>y<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(x_2, y_2)<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord\"><span class=\"mord mathnormal\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li data-start=\"452\" data-end=\"480\">\n<p data-start=\"454\" data-end=\"480\">Target point:<\/p>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(x, y)<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/li>\n<\/ul>\n<h3 data-start=\"487\" data-end=\"506\"><strong data-start=\"494\" data-end=\"506\">Formula:<\/strong><\/h3>\n<p data-start=\"1024\" data-end=\"1123\">\n<p><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>y<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>\u22c5<\/mo><mo fence=\"false\" maxsize=\"1.8em\" minsize=\"1.8em\" stretchy=\"true\">[<\/mo><msub><mi>Q<\/mi><mn>11<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>y<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>Q<\/mi><mn>21<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>y<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>Q<\/mi><mn>12<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>y<\/mi><mo>\u2212<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>Q<\/mi><mn>22<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>y<\/mi><mo>\u2212<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo fence=\"false\" maxsize=\"1.8em\" minsize=\"1.8em\" stretchy=\"true\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x, y) = \\frac{1}{(x_2 &#8211; x_1)(y_2 &#8211; y_1)} \\cdot \\Big[ Q_{11} (x_2 &#8211; x)(y_2 &#8211; y) + Q_{21} (x &#8211; x_1)(y_2 &#8211; y) + Q_{12} (x_2 &#8211; x)(y &#8211; y_1) + Q_{22} (x &#8211; x_1)(y &#8211; y_1) \\Big]<\/annotation><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<h3 data-start=\"695\" data-end=\"751\"><strong data-start=\"702\" data-end=\"751\">Simplified Step-by-Step (Used in Calculator):<\/strong><\/h3>\n<ol data-start=\"753\" data-end=\"824\">\n<li data-start=\"753\" data-end=\"824\">\n<p data-start=\"756\" data-end=\"824\"><strong data-start=\"756\" data-end=\"786\">Interpolate in x direction<\/strong> (at<\/p>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">y = y_1<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span> and<\/p>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><msub><mi>y<\/mi><mn>2<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">y = y_2<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>):<\/li>\n<\/ol>\n<p data-start=\"1024\" data-end=\"1123\">\n<p><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>f<\/mi><mrow><mi>x<\/mi><mi>y<\/mi><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>\u22c5<\/mo><msub><mi>Q<\/mi><mn>11<\/mn><\/msub><mo>+<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>\u22c5<\/mo><msub><mi>Q<\/mi><mn>21<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">f_{xy1} = \\frac{(x_2 &#8211; x)}{(x_2 &#8211; x_1)} \\cdot Q_{11} + \\frac{(x &#8211; x_1)}{(x_2 &#8211; x_1)} \\cdot Q_{21}<\/annotation><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<p data-start=\"1024\" data-end=\"1123\">\n<p><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>f<\/mi><mrow><mi>x<\/mi><mi>y<\/mi><mn>2<\/mn><\/mrow><\/msub><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>\u22c5<\/mo><msub><mi>Q<\/mi><mn>12<\/mn><\/msub><mo>+<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>\u22c5<\/mo><msub><mi>Q<\/mi><mn>22<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">f_{xy2} = \\frac{(x_2 &#8211; x)}{(x_2 &#8211; x_1)} \\cdot Q_{12} + \\frac{(x &#8211; x_1)}{(x_2 &#8211; x_1)} \\cdot Q_{22}<\/annotation><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<ol start=\"2\" data-start=\"1036\" data-end=\"1075\">\n<li data-start=\"1036\" data-end=\"1075\">\n<p data-start=\"1039\" data-end=\"1075\"><strong data-start=\"1039\" data-end=\"1075\">Then interpolate in y direction:<\/strong><\/p>\n<\/li>\n<\/ol>\n<p data-start=\"1024\" data-end=\"1123\">\n<p><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>y<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>y<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>\u22c5<\/mo><msub><mi>f<\/mi><mrow><mi>x<\/mi><mi>y<\/mi><mn>1<\/mn><\/mrow><\/msub><mo>+<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mi>y<\/mi><mo>\u2212<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>y<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>\u22c5<\/mo><msub><mi>f<\/mi><mrow><mi>x<\/mi><mi>y<\/mi><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/semantics><\/math><\/p>\n<p>&nbsp;<\/p>\n<h3 data-start=\"1189\" data-end=\"1205\"><strong data-start=\"1196\" data-end=\"1205\">Note:<\/strong><\/h3>\n<ul data-start=\"1206\" data-end=\"1363\">\n<li data-start=\"1206\" data-end=\"1283\">\n<p data-start=\"1208\" data-end=\"1283\">This method assumes the point<\/p>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(x, y)<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mclose\">)<\/span><\/span><\/span> lies inside or near the rectangle.<\/li>\n<li data-start=\"1284\" data-end=\"1363\">\n<p data-start=\"1286\" data-end=\"1363\">If<\/p>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(x_1 = x_2)<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span> or<\/p>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><msub><mi>y<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(y_1 = y_2)<\/annotation><\/semantics><\/math><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span>, the formula breaks (division by zero).<\/li>\n<\/ul>\n<\/div><\/div><\/div><\/div><div class=\"vc_column tdi_11  wpb_column vc_column_container tdc-column td-pb-span4 td-is-sticky\">\n<style scoped>\n\n\/* custom css *\/\n.tdi_11{\r\n                    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a{\r\n\t\t\t\t\tdisplay: flex;\r\n                  \tmargin-left: 0;\r\n                }.td_block_list_menu .td-blm-menu-item-txt{\r\n\t\t\t\t\tdisplay: flex;\r\n\t\t\t\t\talign-items: center;\r\n\t\t\t\t\tflex-grow: 1;\r\n\t\t\t\t}.td_block_list_menu .sub-menu{\r\n                  padding-left: 22px;\r\n                }.td_block_list_menu .sub-menu li{\r\n                  font-size: 13px;\r\n                }.td_block_list_menu li.current-menu-item > a,\r\n\t\t\t\t.td_block_list_menu li.current-menu-ancestor > a,\r\n\t\t\t\t.td_block_list_menu li.current-category-ancestor > a{\r\n\t\t\t\t    color: #4db2ec;\r\n\t\t\t\t}.td_block_list_menu .td-blm-sub-icon{\r\n\t\t\t\t\tdisplay: flex;\r\n\t\t\t\t\talign-items: center;\r\n\t\t\t\t\tjustify-content: center;\r\n\t\t\t\t\tmargin-left: .6em;\r\n\t\t\t\t\tpadding: 0 .6em;\r\n\t\t\t\t\ttransition: transform .2s ease-in-out;\r\n\t\t\t\t}.td_block_list_menu .td-blm-sub-icon svg{\r\n\t\t\t\t\tdisplay: block;\r\n\t\t\t\t\twidth: 1em;\r\n\t\t\t\t\theight: auto;\r\n\t\t\t\t}.td_block_list_menu .td-blm-sub-icon svg,\r\n\t\t\t\t.td_block_list_menu .td-blm-sub-icon svg *{\r\n\t\t\t\t\tfill: currentColor;\r\n\t\t\t\t}.td_block_list_menu.td-blm-display-accordion .menu-item-has-children ul{\r\n\t\t\t\t\tdisplay: none;\r\n\t\t\t\t}.td_block_list_menu.td-blm-display-accordion .menu-item-has-children-open > a > .td-blm-sub-icon{\r\n\t\t\t\t\ttransform: rotate(180deg);\r\n\t\t\t\t}.td_block_list_menu.td-blm-display-horizontal ul{\r\n\t\t\t\t\tdisplay: flex;\r\n\t\t\t\t}body .tdi_12 ul{\r\n\t\t\t\t\ttext-align: left;\r\n\t\t\t\t\tjustify-content: flex-start;\r\n\t\t\t\t}body .tdi_12 ul li a{\r\n\t\t\t\t\tjustify-content: flex-start;\r\n\t\t\t\t}body .tdi_12 .td-blm-menu-item-txt{\r\n\t\t\t\t\tflex-grow: 1;\r\n\t\t\t\t}\n<\/style><div class=\"td-block-title-wrap\"><h4 class=\"block-title td-block-title\"><span class=\"td-pulldown-size\">Math Calculators<\/span><\/h4><\/div><div id=tdi_12 class=\"td_block_inner td-fix-index\"><div class=\"menu-math-container\"><ul id=\"menu-math\" class=\"menu\"><li id=\"menu-item-3614\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-3614\"><a href=\"https:\/\/www.ezcalculatoronline.com\/math\/average-percentage-calculator\/\"><span class=\"td-blm-menu-item-txt\">Average Percentage Calculator<\/span><\/a><\/li>\n<li id=\"menu-item-3615\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-3615\"><a href=\"https:\/\/www.ezcalculatoronline.com\/math\/fraction-to-percent-calculator\/\"><span class=\"td-blm-menu-item-txt\">Fraction to Percent Calculator<\/span><\/a><\/li>\n<li id=\"menu-item-6528\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-6528\"><a href=\"https:\/\/www.ezcalculatoronline.com\/math\/half-life-calculator\/\"><span class=\"td-blm-menu-item-txt\">Half-Life Calculator<\/span><\/a><\/li>\n<li id=\"menu-item-6529\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-6529\"><a href=\"https:\/\/www.ezcalculatoronline.com\/math\/bilinear-interpolation-calculator\/\"><span class=\"td-blm-menu-item-txt\">Bilinear Interpolation Calculator<\/span><\/a><\/li>\n<li id=\"menu-item-4222\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-4222\"><a href=\"https:\/\/www.ezcalculatoronline.com\/math\/multiplicative-inverse-calculator\/\"><span class=\"td-blm-menu-item-txt\">Multiplicative Inverse Calculator<\/span><\/a><\/li>\n<li id=\"menu-item-6534\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-6534\"><a href=\"https:\/\/www.ezcalculatoronline.com\/math\/scientific-notation-calculator\/\"><span class=\"td-blm-menu-item-txt\">Scientific Notation Calculator<\/span><\/a><\/li>\n<li id=\"menu-item-6530\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-6530\"><a href=\"https:\/\/www.ezcalculatoronline.com\/math\/slope-percentage-calculator\/\"><span class=\"td-blm-menu-item-txt\">Slope Percentage Calculator<\/span><\/a><\/li>\n<li id=\"menu-item-6532\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-6532\"><a href=\"https:\/\/www.ezcalculatoronline.com\/math\/greatest-common-factor-calculator\/\"><span class=\"td-blm-menu-item-txt\">Greatest Common Factor Calculator<\/span><\/a><\/li>\n<li id=\"menu-item-6533\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-6533\"><a href=\"https:\/\/www.ezcalculatoronline.com\/math\/rounding-calculator\/\"><span class=\"td-blm-menu-item-txt\">Rounding Calculator<\/span><\/a><\/li>\n<li id=\"menu-item-6531\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-6531\"><a href=\"https:\/\/www.ezcalculatoronline.com\/math\/slope-calculator\/\"><span class=\"td-blm-menu-item-txt\">Slope Calculator<\/span><\/a><\/li>\n<li id=\"menu-item-3617\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-3617\"><a href=\"https:\/\/www.ezcalculatoronline.com\/games\/math-practice\/\"><span class=\"td-blm-menu-item-txt\">Math Practice<\/span><\/a><\/li>\n<li id=\"menu-item-3618\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-3618\"><a href=\"https:\/\/www.ezcalculatoronline.com\/math\/cubic-equation-calculator\/\"><span class=\"td-blm-menu-item-txt\">Cubic Equation Calculator<\/span><\/a><\/li>\n<li id=\"menu-item-7640\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-7640\"><a href=\"https:\/\/www.ezcalculatoronline.com\/math\/root-calculator\/\"><span class=\"td-blm-menu-item-txt\">Root Calculator<\/span><\/a><\/li>\n<li id=\"menu-item-4221\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-4221\"><a href=\"https:\/\/www.ezcalculatoronline.com\/math\/quadratic-formula-calculator\/\"><span class=\"td-blm-menu-item-txt\">Quadratic Formula Calculator<\/span><\/a><\/li>\n<li id=\"menu-item-3616\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-3616\"><a href=\"https:\/\/www.ezcalculatoronline.com\/games\/multiplication-tables-practice\/\"><span class=\"td-blm-menu-item-txt\">Multiplication Tables Practice<\/span><\/a><\/li>\n<li id=\"menu-item-4223\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-4223\"><a href=\"https:\/\/www.ezcalculatoronline.com\/math\/average-rate-of-change-calculator\/\"><span class=\"td-blm-menu-item-txt\">Average Rate of Change Calculator<\/span><\/a><\/li>\n<\/ul><\/div><\/div><\/div><div class=\"wpb_wrapper td_block_wrap vc_raw_html tdi_14 \"><div class=\"td-fix-index\"><script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-7396755811986582\"\n     crossorigin=\"anonymous\"><\/script>\n<!-- ads 300 x 250 -->\n<ins class=\"adsbygoogle\"\n     style=\"display:inline-block;width:300px;height:250px\"\n     data-ad-client=\"ca-pub-7396755811986582\"\n     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[]).push({});\n<\/script><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div>","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":4104,"parent":3595,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-4093","page","type-page","status-publish","has-post-thumbnail"],"rankMath":{"parentDomain":"ezcalculatoronline.com","noFollowDomains":[],"noFollowExcludeDomains":[],"noFollowExternalLinks":false,"featuredImageNotice":"The featured image should be at least 200 by 200 pixels to be picked up by Facebook and other social media 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