قاعدة كرامر لحل نظام من معادلتين

في هذا الدرس سنتعلم **طريقة كرامر** لحل نظام المعادلات الخطية.

ما هي طريقة كرامر؟

تعتمد طريقة كرامر على حساب **محددات المصفوفات** المستخرجة من نظام المعادلات.

ومن أهم مميزاتها أنها طريقة منتظمة، لذلك يمكن تحويلها بسهولة إلى خوارزميات تنفذها أجهزة الحاسب.

مثال

لدينا نظام المعادلتين:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="5x-6y=15" data-tex="5x-6y=15" style="display:block;margin:0 auto;vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="12.71ex" height="1.971ex" role="img" focusable="false" viewBox="0 -666 5618 871" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="35" xlink:href="#MJXG-TEX-N-35"></use></g><g data-mml-node="mi" transform="translate(500,0)"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1294.2,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(2294.4,0)"><use data-c="36" xlink:href="#MJXG-TEX-N-36"></use></g><g data-mml-node="mi" transform="translate(2794.4,0)"><use data-c="1D466" xlink:href="#MJXG-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(3562.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(4618,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="35" xlink:href="#MJXG-TEX-N-35" transform="translate(500,0)"></use></g></g></g></svg></div>

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="3x+4y=-29" data-tex="3x+4y=-29" style="display:block;margin:0 auto;vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="14.471ex" height="1.995ex" role="img" focusable="false" viewBox="0 -677 6396 882" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mi" transform="translate(500,0)"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1294.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2294.4,0)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(2794.4,0)"><use data-c="1D466" xlink:href="#MJXG-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(3562.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(4618,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(5396,0)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use><use data-c="39" xlink:href="#MJXG-TEX-N-39" transform="translate(500,0)"></use></g></g></g></svg></div>

الخطوة الأولى: تكوين مصفوفة المعاملات

نأخذ معاملات المتغيرين فقط، فنحصل على مصفوفة المعاملات:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="\begin{bmatrix} 5 &amp; -6\\ 3 &amp; 4 \end{bmatrix}" data-tex="\begin{bmatrix} 5 &amp; -6\\ 3 &amp; 4 \end{bmatrix}" style="display:block;margin:0 auto;vertical-align: -2.149ex;" xmlns="http://www.w3.org/2000/svg" width="8.674ex" height="5.43ex" role="img" focusable="false" viewBox="0 -1450 3834 2400" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="5B" xlink:href="#MJXG-TEX-S3-5B"></use></g><g data-mml-node="mtable" transform="translate(528,0)"><g data-mml-node="mtr" transform="translate(0,700)"><g data-mml-node="mtd"><g data-mml-node="mn"><use data-c="35" xlink:href="#MJXG-TEX-N-35"></use></g></g><g data-mml-node="mtd" transform="translate(1500,0)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="36" xlink:href="#MJXG-TEX-N-36"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-700)"><g data-mml-node="mtd"><g data-mml-node="mn"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g></g><g data-mml-node="mtd" transform="translate(1889,0)"><g data-mml-node="mn"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g></g></g></g><g data-mml-node="mo" transform="translate(3306,0) translate(0 -0.5)"><use data-c="5D" xlink:href="#MJXG-TEX-S3-5D"></use></g></g></g></g></svg></div>

ثم نحسب محددها بالطريقة التي تعلمناها سابقًا:

**حاصل ضرب القطر الرئيسي ناقص حاصل ضرب القطر الثانوي.**

فتكون قيمة محدد المعاملات:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="D=38" data-tex="D=38" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.153ex" height="1.731ex" role="img" focusable="false" viewBox="0 -683 3161.6 765" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g><g data-mml-node="mo" transform="translate(1105.8,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2161.6,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use><use data-c="38" xlink:href="#MJXG-TEX-N-38" transform="translate(500,0)"></use></g></g></g></svg></div>

الخطوة الثانية: إيجاد قيمة <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="x" data-tex="x" style="display:inline-block;vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g></g></g></svg></span>

لحساب محدد <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="D_x" data-tex="D_x" style="display:inline-block;vertical-align: -0.357ex;" xmlns="http://www.w3.org/2000/svg" width="2.976ex" height="1.902ex" role="img" focusable="false" viewBox="0 -683 1315.5 840.8" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g><g data-mml-node="mi" transform="translate(861,-150) scale(0.707)"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g></g></g></g></svg></span>:

نستبدل **عمود <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="x" data-tex="x" style="display:inline-block;vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g></g></g></svg></span>** بالحدود الثابتة الموجودة بعد علامة المساواة، بينما يبقى عمود <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="y" data-tex="y" style="display:inline-block;vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJXG-TEX-I-1D466"></use></g></g></g></svg></span> كما هو.

بعد حساب المحدد نجد:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="D_x=-114" data-tex="D_x=-114" style="display:block;margin:0 auto;vertical-align: -0.357ex;" xmlns="http://www.w3.org/2000/svg" width="11.147ex" height="1.902ex" role="img" focusable="false" viewBox="0 -683 4927 840.8" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g><g data-mml-node="mi" transform="translate(861,-150) scale(0.707)"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g></g><g data-mml-node="mo" transform="translate(1593.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(2649,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(3427,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="31" xlink:href="#MJXG-TEX-N-31" transform="translate(500,0)"></use><use data-c="34" xlink:href="#MJXG-TEX-N-34" transform="translate(1000,0)"></use></g></g></g></svg></div>

ثم نطبق قانون كرامر:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="x=\frac{D_x}{D} =\frac{-114}{38} =-3" data-tex="x=\frac{D_x}{D} =\frac{-114}{38} =-3" style="display:block;margin:0 auto;vertical-align: -1.602ex;" xmlns="http://www.w3.org/2000/svg" width="23.358ex" height="4.676ex" role="img" focusable="false" viewBox="0 -1359 10324.1 2067" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(849.8,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(1905.6,0)"><g data-mml-node="msub" transform="translate(220,676)"><g data-mml-node="mi"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g><g data-mml-node="mi" transform="translate(861,-150) scale(0.707)"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g></g><g data-mml-node="mi" transform="translate(463.7,-686)"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g><rect width="1515.5" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(3938.8,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(4994.6,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="31" xlink:href="#MJXG-TEX-N-31" transform="translate(500,0)"></use><use data-c="34" xlink:href="#MJXG-TEX-N-34" transform="translate(1000,0)"></use></g></g><g data-mml-node="mn" transform="translate(859,-686)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use><use data-c="38" xlink:href="#MJXG-TEX-N-38" transform="translate(500,0)"></use></g><rect width="2478" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(7990.4,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(9046.1,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(9824.1,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g></g></g></svg></div>

الخطوة الثالثة: إيجاد قيمة <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="y" data-tex="y" style="display:inline-block;vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJXG-TEX-I-1D466"></use></g></g></g></svg></span>

لحساب محدد <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="D_y" data-tex="D_y" style="display:inline-block;vertical-align: -0.667ex;" xmlns="http://www.w3.org/2000/svg" width="2.845ex" height="2.213ex" role="img" focusable="false" viewBox="0 -683 1257.5 978" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g><g data-mml-node="mi" transform="translate(861,-150) scale(0.707)"><use data-c="1D466" xlink:href="#MJXG-TEX-I-1D466"></use></g></g></g></g></svg></span>:

نستبدل **عمود <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="y" data-tex="y" style="display:inline-block;vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJXG-TEX-I-1D466"></use></g></g></g></svg></span>** بالحدود الثابتة، بينما يبقى عمود <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="x" data-tex="x" style="display:inline-block;vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g></g></g></svg></span> كما هو.

بعد حساب المحدد نجد:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="D_y=-190" data-tex="D_y=-190" style="display:block;margin:0 auto;vertical-align: -0.667ex;" xmlns="http://www.w3.org/2000/svg" width="11.016ex" height="2.213ex" role="img" focusable="false" viewBox="0 -683 4869 978" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g><g data-mml-node="mi" transform="translate(861,-150) scale(0.707)"><use data-c="1D466" xlink:href="#MJXG-TEX-I-1D466"></use></g></g><g data-mml-node="mo" transform="translate(1535.3,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(2591,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(3369,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="39" xlink:href="#MJXG-TEX-N-39" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJXG-TEX-N-30" transform="translate(1000,0)"></use></g></g></g></svg></div>

ثم نطبق القانون:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="y=\frac{D_y}{D} =\frac{-190}{38} =-5" data-tex="y=\frac{D_y}{D} =\frac{-190}{38} =-5" style="display:block;margin:0 auto;vertical-align: -1.602ex;" xmlns="http://www.w3.org/2000/svg" width="23.041ex" height="4.855ex" role="img" focusable="false" viewBox="0 -1438 10184.1 2146" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJXG-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(767.8,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(1823.6,0)"><g data-mml-node="msub" transform="translate(220,755)"><g data-mml-node="mi"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g><g data-mml-node="mi" transform="translate(861,-150) scale(0.707)"><use data-c="1D466" xlink:href="#MJXG-TEX-I-1D466"></use></g></g><g data-mml-node="mi" transform="translate(434.7,-686)"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g><rect width="1457.5" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(3798.8,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(4854.6,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="39" xlink:href="#MJXG-TEX-N-39" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJXG-TEX-N-30" transform="translate(1000,0)"></use></g></g><g data-mml-node="mn" transform="translate(859,-686)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use><use data-c="38" xlink:href="#MJXG-TEX-N-38" transform="translate(500,0)"></use></g><rect width="2478" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(7850.4,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(8906.1,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(9684.1,0)"><use data-c="35" xlink:href="#MJXG-TEX-N-35"></use></g></g></g></svg></div>

الخلاصة

لحل نظام معادلتين بطريقة كرامر:

1. نكوّن مصفوفة المعاملات ونحسب محددها <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="D" data-tex="D" style="display:inline-block;vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.873ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 828 683" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g></g></g></svg></span>. 2. نحسب محدد <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="D_x" data-tex="D_x" style="display:inline-block;vertical-align: -0.357ex;" xmlns="http://www.w3.org/2000/svg" width="2.976ex" height="1.902ex" role="img" focusable="false" viewBox="0 -683 1315.5 840.8" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g><g data-mml-node="mi" transform="translate(861,-150) scale(0.707)"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g></g></g></g></svg></span> باستبدال عمود <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="x" data-tex="x" style="display:inline-block;vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g></g></g></svg></span> بالحدود الثابتة. 3. نحسب محدد <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="D_y" data-tex="D_y" style="display:inline-block;vertical-align: -0.667ex;" xmlns="http://www.w3.org/2000/svg" width="2.845ex" height="2.213ex" role="img" focusable="false" viewBox="0 -683 1257.5 978" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g><g data-mml-node="mi" transform="translate(861,-150) scale(0.707)"><use data-c="1D466" xlink:href="#MJXG-TEX-I-1D466"></use></g></g></g></g></svg></span> باستبدال عمود <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="y" data-tex="y" style="display:inline-block;vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJXG-TEX-I-1D466"></use></g></g></g></svg></span> بالحدود الثابتة. 4. نحسب الحل باستخدام:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="x=\frac{D_x}{D}" data-tex="x=\frac{D_x}{D}" style="display:block;margin:0 auto;vertical-align: -1.552ex;" xmlns="http://www.w3.org/2000/svg" width="8.283ex" height="4.627ex" role="img" focusable="false" viewBox="0 -1359 3661 2045" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(849.8,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(1905.6,0)"><g data-mml-node="msub" transform="translate(220,676)"><g data-mml-node="mi"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g><g data-mml-node="mi" transform="translate(861,-150) scale(0.707)"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g></g><g data-mml-node="mi" transform="translate(463.7,-686)"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g><rect width="1515.5" height="60" x="120" y="220"></rect></g></g></g></svg></div>

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="y=\frac{D_y}{D}" data-tex="y=\frac{D_y}{D}" style="display:block;margin:0 auto;vertical-align: -1.552ex;" xmlns="http://www.w3.org/2000/svg" width="7.966ex" height="4.805ex" role="img" focusable="false" viewBox="0 -1438 3521 2124" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJXG-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(767.8,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(1823.6,0)"><g data-mml-node="msub" transform="translate(220,755)"><g data-mml-node="mi"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g><g data-mml-node="mi" transform="translate(861,-150) scale(0.707)"><use data-c="1D466" xlink:href="#MJXG-TEX-I-1D466"></use></g></g><g data-mml-node="mi" transform="translate(434.7,-686)"><use data-c="1D437" xlink:href="#MJXG-TEX-I-1D437"></use></g><rect width="1457.5" height="60" x="120" y="220"></rect></g></g></g></svg></div>

وبذلك نحصل على قيم المتغيرين بطريقة منظمة وسهلة.