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<!DOCTYPE html>
<html data-require="math math-format expressions">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Multi-step equations with distribution</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="summary">
Solving linear equations with distribution and combining like terms.
The coefficient to be distributed is chosen thusly:
- distribute positive numbers 22.5% of the time
- distribute a minus sign 40% of the time
- distribute other negative numbers 37.5% of the time
Problem types:
prob1 A = Bx + C(Dx + E)
prob2 A = B(C + Dx) + Ex
prob3 A(Bx + C) = D(E − Fx) + Gx
prob4 Ax + B(Cx + D) = Ex + F
prob5 A + Bx = C(Dx + E)
prob6 Ax + B(C + Dx) = E
prob7 A(Bx + C) = Dx + E
prob8 A(B + Cx) + D = E + Fx
prob9 A(Bx + C) = D(E + Fx)
</div>
<div class="vars">
<var id="X">randVar()</var>
</div>
<div class="problems">
<div id="prob1"> <!-- A = Bx + C(Dx + E) -->
<div class="vars" data-ensure="(B + C * D !== 0) &&
(A - C * E !== 0) && (B + C * D !== 1)">
<var id="A">randRangeNonZero(-10, 10)</var>
<var id="B">randRangeNonZero(-10, 10)</var>
<var id="C">
randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1])
</var>
<var id="D">randRangeNonZero(-10, 10)</var>
<var id="E">randRange(1, 3)</var>
<var id="SOLUTION">(A - C * E) / (B + C * D)</var>
</div>
<p class="question">Solve for <code><var>X</var></code>:</p>
<p class="problem"><code>\qquad
<var>A</var> = <var>expr(["+", ["*", B, X], ["*", C, ["+",
["*", D, X], E]]])</var>
</code></p>
<div class="solution" data-type="multiple">
<p>
<code><var>X</var> =</code>
<span class="sol"><var>SOLUTION</var></span>
</p>
</div>
<div class="hints">
<p>
Try simplifying the right side of the equation before
solving it.
</p>
<div>
<p data-if="C === -1">
Distribute the negative in front of the parentheses.
Be careful! The negative sign in front of the
parentheses means we're multiplying by
<code>\pink{-1}</code>:
</p><p data-else-if="C < 0">
Distribute the <code>\pink{<var>C</var>}</code>.
Be careful to pay attention to the negative sign
when you distribute:
</p><p data-else>
Distribute the <code>\pink{<var>C</var>}</code>:
</p>
<p><code>\qquad\begin{eqnarray}
<var>A</var> &amp;=&amp; <var>expr(["*", B, X])</var> +
\pink{<var>C</var>}\blue{(<var>expr(["+", ["*", D, X], E])</var>)} \\ \\
<var>A</var> &amp;=&amp; <var>expr(["*", B, X])</var> +
\pink{(<var>C</var>)}\blue{(<var>expr(["*", D, X])</var>)} +
\pink{(<var>C</var>)}\blue{(<var>E</var>)}
\end{eqnarray}
</code></p>
</div>
<div>
<p>Multiply:</p>
<p><code>\qquad
<var>A</var> =
<var>expr(["+", ["*", B, X], ["*", C * D, X], C * E])</var>
</code></p>
</div>
<div>
<p>
Combine the <span class="hint_blue">
<code><var>X</var></code> terms</span>:
</p>
<p><code>\qquad\begin{eqnarray}
<var>A</var> &amp;=&amp;
\blue{<var>expr(["+", ["*", B, X], ["*", C * D, X]])</var>} + <var>C * E</var> \\ \\
<var>A</var> &amp;=&amp;
\blue{<var>expr(["*", B + C * D, X])</var>} + <var>C * E</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
<var>C * E &lt; 0 ? "Add" : "Subtract"</var>
<code>\green{<var>abs(C * E)</var>}</code>
<var>C * E &lt; 0 ? "to" : "from"</var> both sides
to isolate the <code><var>X</var></code> term on the
right side:
</p>
<p><code>\qquad\begin{eqnarray}
<var>A</var> &amp;=&amp;
<var>expr(["*", B + C * D, X])</var> \green{{} + <var>C * E</var>} \\ \\
\green{{}+<var>-C * E</var>} &amp;&amp;
\green{{}+<var>-C * E</var>} \\ \\
<var>A - C * E</var> &amp;=&amp;
<var>expr(["*", B + C * D, X])</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Divide both sides by
<code>\green{<var>B + C * D</var>}</code>
to isolate <code><var>X</var></code>:
</p>
<p><code>\qquad\begin{eqnarray}
<var>A - C * E</var> &amp;=&amp;
<var>expr(["*", B + C * D, X])</var> \\ \\
\dfrac{<var>A - C * E</var>}
{\green{<var>B + C * D</var>}} &amp;=&amp;
\dfrac{\green{\cancel{<var>B + C * D</var>}}
<var>X</var>}{\green{\cancel{<var>B + C * D</var>}}}
\end{eqnarray}
</code></p>
</div>
<div>
<p>Simplify:</p>
<p><code>\qquad
<var>fractionReduce(A - C * E, B + C * D)</var>
= <var>X</var>
</code></p>
</div>
</div>
</div>
<div id="prob2"> <!-- A = B(C + Dx) + Ex -->
<div class="vars" data-ensure="(B * D + E !== 0) &&
(A - B * C) && (B * D + E)">
<var id="A">randRangeNonZero(-10, 10)</var>
<var id="B">
randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1])
</var>
<var id="C">randRange(1, 3)</var>
<var id="D">randRangeNonZero(-10, 10)</var>
<var id="E">randRangeNonZero(-10, 10)</var>
<var id="SOLUTION">(A - B * C) / (B * D + E)</var>
</div>
<p class="question">Solve for <code><var>X</var></code>:</p>
<p class="problem"><code>\qquad
<var>A</var> = <var>expr(["+", ["*", B, ["+", C, ["*", D, X]]],
["*", E, X]])</var>
</code></p>
<div class="solution" data-type="multiple">
<p>
<code><var>X</var> =</code>
<span class="sol"><var>SOLUTION</var></span>
</p>
</div>
<div class="hints">
<p>
Try simplifying the right side of the equation before
solving it.
</p>
<div>
<p data-if="B === -1">
Distribute the negative in front of the parentheses.
Be careful! The negative sign in front of the
parentheses means we're multiplying by
<code>\pink{-1}</code>:
</p><p data-else-if="B < 0">
Distribute the <code>\pink{<var>B</var>}</code>.
Be careful to pay attention to the negative sign
when you distribute:
</p><p data-else>
Distribute the <code>\pink{<var>B</var>}</code>:
</p>
<p><code>\qquad\begin{eqnarray}
<var>A</var> &amp;=&amp;
\pink{<var>B</var>}\blue{(<var>expr(["+", C, ["*", D, X]])</var>)} +
<var>expr(["*", E, X])</var> \\ \\
<var>A</var> &amp;=&amp;
\pink{(<var>B</var>)}\blue{(<var>C</var>)} +
\pink{(<var>B</var>)}\blue{(<var>expr(["*", D, X])</var>)} +
<var>expr(["*", E, X])</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>Multiply:</p>
<p><code>\qquad
<var>A</var> =
<var>expr(["+", B * C, ["*", B * D, X], ["*", E, X]])</var>
</code></p>
</div>
<div>
<p>
Combine the <span class="hint_blue">
<code><var>X</var></code> terms</span>:
</p>
<p><code>\qquad\begin{eqnarray}
<var>A</var> &amp;=&amp;
<var>B * C</var> \blue{{} + <var>expr(["*", B * D, X])</var>}
\blue{{} + <var>expr(["*", E, X])</var>} \\ \\
<var>A</var> &amp;=&amp;
<var>B * C</var> \blue{{} + <var>expr(["*", B * D + E, X])</var>}
\end{eqnarray}
</code></p>
</div>
<div>
<p>
<var>B * C &lt; 0 ? "Add" : "Subtract"</var>
<code>\green{<var>abs(B * C)</var>}</code>
<var>B * C &lt; 0 ? "to" : "from"</var> both sides
to isolate the <code><var>X</var></code> term on the
right side:
</p>
<p><code>\qquad\begin{eqnarray}
<var>A</var> &amp;=&amp;
\green{<var>B * C</var>} + <var>expr(["*", B * D + E, X])</var> \\ \\
\green{{}+<var>-B * C</var>} &amp;&amp;
\green{{}+<var>-B * C</var>} \\ \\
<var>A - B * C</var> &amp;=&amp;
<var>expr(["*", B * D + E, X])</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Divide both sides by
<code>\green{<var>B * D + E</var>}</code>
to isolate <code><var>X</var></code>:
</p>
<p><code>\qquad\begin{eqnarray}
<var>A - B * C</var> &amp;=&amp;
<var>expr(["*", B * D + E, X])</var> \\ \\
\dfrac{<var>A - B * C</var>}
{\green{<var>B * D + E</var>}} &amp;=&amp;
\dfrac{\green{\cancel{<var>B * D + E</var>}}
<var>X</var>}{\green{\cancel{<var>B * D + E</var>}}}
\end{eqnarray}
</code></p>
</div>
<div>
<p>Simplify:</p>
<p><code>\qquad
<var>fractionReduce(A - B * C, B * D + E)</var>
= <var>X</var>
</code></p>
</div>
</div>
</div>
<div id="prob3"> <!-- A(Bx + C) = D(E − Fx) + Gx -->
<div class="vars" data-ensure="D * F + G !== 0 &&
((A * B) - (D * F + G)) !== 0 &&
((A * B) - (D * F + G)) !== 1">
<var id="A">
randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1])
</var>
<var id="B">randRangeNonZero(-10, 10)</var>
<var id="C">randRange(1, 3)</var>
<var id="D">
randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1])
</var>
<var id="E">randRangeNonZero(-10, 10)</var>
<var id="F">randRangeNonZero(-10, 10)</var>
<var id="G">randRangeNonZero(-10, 10)</var>
<var id="SOLUTION">(D * E - A * C) / ((A * B) - (D * F + G))</var>
</div>
<p class="question">Solve for <code><var>X</var></code>:</p>
<p class="problem"><code>\qquad
<var>expr(["*", A, ["+", ["*", B, X], C]])</var> =
<var>expr(["+", ["*", D, ["+", E, ["*", F, X]]],
["*", G, X]])</var>
</code></p>
<div class="solution" data-type="multiple">
<p>
<code><var>X</var> =</code>
<span class="sol"><var>SOLUTION</var></span>
</p>
</div>
<div class="hints">
<p>
Try simplifying each side of the equation before
solving it.
</p>
<div>
<p data-if="A === -1">
Distribute the negative in front of the parentheses.
Be careful! The negative sign in front of the
parentheses means we're multiplying by
<code>\pink{-1}</code>:
</p><p data-else-if="A < 0">
Distribute the <code>\pink{<var>A</var>}</code>.
Be careful to pay attention to the negative sign
when you distribute:
</p><p data-else>
Distribute the <code>\pink{<var>A</var>}</code>:
</p>
<p><code>\qquad\begin{eqnarray}
\pink{<var>A</var>}\blue{(<var>expr(["+", ["*", B, X], C])</var>)} &amp;=&amp;
<var>expr(["+", ["*", D, ["+", E, ["*", F, X]]], ["*", G, X]])</var> \\ \\
\pink{(<var>A</var>)}\blue{(<var>expr(["*", B, X])</var>)} +
\pink{(<var>A</var>)}\blue{(<var>C</var>)} &amp;=&amp;
<var>expr(["+", ["*", D, ["+", E, ["*", F, X]]], ["*", G, X]])</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>Multiply:</p>
<p><code>\qquad
<var>expr(["+", ["*", A * B, X], A * C])</var> =
<var>expr(["+", ["*", D, ["+", E, ["*", F, X]]], ["*", G, X]])</var>
</code></p>
</div>
<div>
<p data-if="D === -1">
Distribute the negative in front of the parentheses.
Be careful! The negative sign in front of the
parentheses means we're multiplying by
<code>\pink{-1}</code>:
</p><p data-else-if="D < 0">
Distribute the <code>\pink{<var>D</var>}</code>.
Be careful to pay attention to the negative sign
when you distribute:
</p><p data-else>
Distribute the <code>\pink{<var>D</var>}</code>:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["+", ["*", A * B, X], A * C])</var> &amp;=&amp;
\pink{(<var>D</var>)}\blue{(<var>expr(["+", E, ["*", F, X]])</var>)} +
<var>expr(["*", G, X])</var> \\ \\
<var>expr(["+", ["*", A * B, X], A * C])</var> &amp;=&amp;
\pink{(<var>D</var>)} \blue{(<var>E</var>)} +
\pink{(<var>D</var>)}
\blue{(<var>expr(["*", F, X])</var>)} +
<var>expr(["*", G, X])</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>Multiply:</p>
<p><code>\qquad
<var>expr(["+", ["*", A * B, X], A * C])</var> =
<var>expr(["+", D * E, ["*", D * F, X], ["*", G, X]])</var>
</code></p>
</div>
<div>
<p>
Combine the <span class="hint_blue">
<code><var>X</var></code> terms</span>:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["+", ["*", A * B, X], A * C])</var> &amp;=&amp;
<var>D * E</var> \blue{{} + <var>expr(["+", ["*", D * F, X], ["*", G, X]])</var>} \\ \\
<var>expr(["+", ["*", A * B, X], A * C])</var> &amp;=&amp;
<var>D * E</var> \blue{{} + <var>expr(["*", D * F + G, X])</var>}
\end{eqnarray}
</code></p>
</div>
<div>
<p>
<var>D * F + G &lt; 0 ? "Add" : "Subtract"</var>
<code>\green{<var>expr(["*", abs(D * F + G), X])</var>}
</code> <var>D * F + G &lt; 0 ? "to" : "from"</var> both
sides to eliminate the <code><var>X</var></code> term
from the right side:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["+", ["*", A * B, X], A * C])</var> &amp;=&amp;
<var>D * E</var> \green{{} + <var>expr(["*", D * F + G, X])</var>} \\ \\
\green{{}+<var>expr(["*", -(D * F + G), X])</var>}
&amp;&amp; \green{{} +
<var>expr(["*", -(D * F + G), X])</var>} \\ \\
<var>expr(["+", ["*", A * B, X], A * C])</var>
\green{+<var>expr(["*", -(D * F + G), X])</var>} &amp;=&amp;
<var>D * E</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Combine the <span class="hint_blue">
<code><var>X</var></code> terms</span>:
</p>
<p><code>\qquad\begin{eqnarray}
\blue{<var>expr(["*", A * B, X])</var>} + <var>A * C</var>
\blue{+<var>expr(["*", -(D * F + G), X])</var>} &amp;=&amp;
<var>D * E</var> \\ \\
\blue{<var>expr(["*", (A * B) - (D * F + G), X])</var>} + <var>A * C</var> &amp;=&amp;
<var>D * E</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
<var>A * C &lt; 0 ? "Add" : "Subtract"</var>
<code>\green{<var>abs(A * C)</var>}</code>
<var>A * C &lt; 0 ? "to" : "from"</var> both sides
to isolate the <code><var>X</var></code> term on the
left side:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["*", (A * B) - (D * F + G), X])</var> \green{{} + <var>A * C</var>} &amp;=&amp;
<var>D * E</var> \\ \\
\green{{}+<var>-A * C</var>} &amp;&amp;
\green{{}+<var>-A * C</var>} \\ \\
<var>expr(["*", (A * B) - (D * F + G), X])</var>
&amp;=&amp; <var>D * E - A * C</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Divide both sides by
<code>\green{<var>(A * B) - (D * F + G)</var>}</code>
to isolate <code><var>X</var></code>:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["*", (A * B) - (D * F + G), X])</var>
&amp;=&amp; <var>D * E - A * C</var> \\ \\
\dfrac{\green{\cancel{<var>(A * B) - (D * F + G)</var>}}
<var>X</var>}{\green{\cancel{<var>(A * B) - (D * F + G)</var>}}}
&amp;=&amp; \dfrac{<var>D * E - A * C</var>}
{\green{<var>(A * B) - (D * F + G)</var>}} \\ \\
\end{eqnarray}
</code></p>
</div>
<div>
<p>Simplify:</p>
<p><code>\qquad
<var>X</var>
= <var>fractionReduce(D * E - A * C, (A * B) - (D * F + G))</var>
</code></p>
</div>
</div>
</div>
<div id="prob4"> <!-- Ax + B(Cx + D) = Ex + F -->
<div class="vars" data-ensure="(F - B * D !== 0) &&
(A + B * C - E !== 0) &&
(A + B * C - E !== 1)">
<var id="A">randRangeNonZero(-8, 8)</var>
<var id="B">
randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1])
</var>
<var id="C">randRange(1, 3)</var>
<var id="D">randRangeNonZero(-10, 10)</var>
<var id="E">randRangeNonZero(-6, 6)</var>
<var id="F">randRangeNonZero(-10, 10)</var>
<var id="SOLUTION">(F - B * D) / (A + B * C - E)</var>
</div>
<p class="question">Solve for <code><var>X</var></code>:</p>
<p class="problem"><code>\qquad
<var>expr(["+", ["*", A, X], ["*", B, ["+",
["*", C, X], D]]])</var> =
<var>expr(["+", ["*", E, X], F])</var>
</code></p>
<div class="solution" data-type="multiple">
<p>
<code><var>X</var> =</code>
<span class="sol"><var>SOLUTION</var></span>
</p>
</div>
<div class="hints">
<p>
Try simplifying the left side of the equation before
solving it.
</p>
<div>
<p data-if="B === -1">
Distribute the negative in front of the parentheses.
Be careful! The negative sign in front of the
parentheses means we're multiplying by
<code>\pink{-1}</code>:
</p><p data-else-if="B < 0">
Distribute the <code>\pink{<var>B</var>}</code>.
Be careful to pay attention to the negative sign
when you distribute:
</p><p data-else>
Distribute the <code>\pink{<var>B</var>}</code>:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["*", A, X])</var> +
\pink{<var>B</var>}
\blue{(<var>expr(["+", ["*", C, X], D])</var>)}
&amp;=&amp;
<var>expr(["+", ["*", E, X], F])</var> \\ \\
<var>expr(["*", A, X])</var> +
\pink{(<var>B</var>)}
\blue{(<var>expr(["*", C, X])</var>)} +
\pink{(<var>B</var>)}
\blue{(<var>D</var>)} &amp;=&amp;
<var>expr(["+", ["*", E, X], F])</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>Multiply:</p>
<p><code>\qquad
<var>expr(["+", ["*", A, X], ["*", B * C, X],
B * D])</var> = <var>expr(["+", ["*", E, X],
F])</var>
</code></p>
</div>
<div>
<p>
<var>E &lt; 0 ? "Add" : "Subtract"</var>
<code>\green{<var>expr(["*", abs(E), X])</var>}
</code> <var>E &lt; 0 ? "to" : "from"</var> both
sides to eliminate the <code><var>X</var></code> term
from the right side:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["+", ["*", A, X], ["*", B * C, X],
B * D])</var> &amp;=&amp;
\green{<var>expr(["*", E, X])</var>} +
<var>F</var> \\ \\
\green{{}+<var>expr(["*", -E, X])</var>}
&amp;=&amp; \green{{} +
<var>expr(["*", -E, X])</var>} \\ \\
<var>expr(["+", ["*", A, X], ["*", B * C, X],
B * D])</var> \green{
+<var>expr(["*", -E, X])</var>} &amp;=&amp;
<var>F</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Combine the <span class="hint_blue">
<code><var>X</var></code> terms</span>:
</p>
<p><code>\qquad\begin{eqnarray}
\blue{<var>expr(["+", ["*", A, X],
["*", B * C, X]])</var>} + <var>B * D</var>
\blue{+<var>expr(["*", -E, X])</var>} &amp;=&amp;
<var>F</var> \\ \\
\blue{<var>expr(["*", A + B * C - E, X])</var>} +
<var>B * D</var> &amp; = &amp; <var>F</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
<var>B * D &lt; 0 ? "Add" : "Subtract"</var>
<code>\green{<var>abs(B * D)</var>}</code>
<var>B * D &lt; 0 ? "to" : "from"</var> both sides
to isolate the <code><var>X</var></code> term on the
left side:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["*", A + B * C - E, X])</var>
\green{+ <var>B * D</var>} &amp; = &amp;
<var>F</var> \\ \\
\green{{}+<var>-B * D</var>} &amp;&amp;
\green{{}+<var>-B * D</var>} \\ \\
<var>expr(["*", A + B * C - E, X])</var>
&amp;=&amp; <var>F - B * D</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Divide both sides by
<code>\green{<var>A + B * C - E</var>}</code>
to isolate <code><var>X</var></code>:
</p>
<p><code>\qquad\begin{eqnarray}
\green{<var>A + B * C - E</var>}<var>X</var> &amp;=&amp;
<var>F - B * D</var> \\ \\
\dfrac{\green{\cancel{<var>A + B * C - E</var>}}
<var>X</var>}{\green{\cancel{<var>A + B * C - E</var>}}}
&amp;=&amp; \dfrac{<var>F - B * D</var>}
{\green{<var>A + B * C - E</var>}} \\ \\
\end{eqnarray}
</code></p>
</div>
<div>
<p>Simplify:</p>
<p><code>\qquad
<var>X</var>
= <var>fractionReduce(F - B * D, A + B * C - E)</var>
</code></p>
</div>
</div>
</div>
<div id="prob5"> <!-- A + Bx = C(Dx + E) -->
<div class="vars" data-ensure="(B - C * D !== 0) && (B - C * D !== 1)">
<var id="A">randRangeNonZero(-8, 8)</var>
<var id="B">randRangeNonZero(-10, 10)</var>
<var id="C">
randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1])
</var>
<var id="D">randRange(1, 3)</var>
<var id="E">randRangeNonZero(-6, 6)</var>
<var id="SOLUTION">(C * E - A) / (B - C * D)</var>
</div>
<p class="question">Solve for <code><var>X</var></code>:</p>
<p class="problem"><code>\qquad
<var>expr(["+", A, ["*", B, X]])</var> =
<var>expr(["*", C, ["+", ["*", D, X], E]])</var>
</code></p>
<div class="solution" data-type="multiple">
<p>
<code><var>X</var> =</code>
<span class="sol"><var>SOLUTION</var></span>
</p>
</div>
<div class="hints">
<p>
Try simplifying the right side of the equation before
solving it.
</p>
<div>
<p data-if="C === -1">
Distribute the negative in front of the parentheses.
Be careful! The negative sign in front of the
parentheses means we're multiplying by
<code>\pink{-1}</code>:
</p><p data-else-if="C < 0">
Distribute the <code>\pink{<var>C</var>}</code>.
Be careful to pay attention to the negative sign
when you distribute:
</p><p data-else>
Distribute the <code>\pink{<var>C</var>}</code>:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["+", A, ["*", B, X]])</var> &amp;=&amp;
\pink{<var>C</var>}\blue{(<var>expr(["+", ["*", D, X], E])</var>)} \\ \\
<var>expr(["+", A, ["*", B, X]])</var> &amp;=&amp;
\pink{(<var>C</var>)}\blue{(<var>expr(["*", D, X])</var>)} +
\pink{(<var>C</var>)}\blue{(<var>E</var>)}
\end{eqnarray}
</code></p>
</div>
<div>
<p>Multiply:</p>
<p><code>\qquad
<var>expr(["+", A, ["*", B, X]])</var> =
<var>expr(["+", ["*", C * D, X], C * E])</var>
</code></p>
</div>
<div>
<p>
<var>C * D &lt; 0 ? "Add" : "Subtract"</var>
<code>\green{<var>expr(["*", abs(C * D), X])</var>}
</code> <var>C * D &lt; 0 ? "to" : "from"</var> both
sides to eliminate the <code><var>X</var></code> term
from the right side:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["+", A, ["*", B, X]])</var> &amp;=&amp;
\green{<var>expr(["*", C * D, X])</var>} + <var>C * E</var> \\ \\
\green{{}+<var>expr(["*", -C * D, X])</var>}
&amp;&amp; \green{{} +
<var>expr(["*", -C * D, X])</var>} \\ \\
<var>expr(["+", A, ["*", B, X]])</var>
\green{{} + <var>expr(["*", -C * D, X])</var>} &amp;=&amp;
<var>C * E</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Combine the <span class="hint_blue">
<code><var>X</var></code> terms</span>:
</p>
<p><code>\qquad\begin{eqnarray}
<var>A</var> \blue{{} + <var>expr(["+", ["*", B, X], ["*", -C * D, X]])</var>}
&amp;=&amp; <var>C * E</var> \\ \\
<var>A</var> \blue{{} + <var>expr(["*", B - C * D, X])</var>} &amp;=&amp; <var>C * E</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
<var>A &lt; 0 ? "Add" : "Subtract"</var>
<code>\green{<var>abs(A)</var>}</code>
<var>A &lt; 0 ? "to" : "from"</var> both sides
to isolate the <code><var>X</var></code> term on the
left side:
</p>
<p><code>\qquad\begin{eqnarray}
\green{<var>A</var>} + <var>expr(["*", B - C * D, X])</var> &amp;=&amp; <var>C * E</var> \\ \\
\green{{}+<var>-A</var>} &amp;&amp;
\green{{}+<var>-A</var>} \\ \\
<var>expr(["*", B - C * D, X])</var> &amp;=&amp; <var>C * E - A</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Divide both sides by
<code>\green{<var>B - C * D</var>}</code>
to isolate <code><var>X</var></code>:
</p>
<p><code>\qquad\begin{eqnarray}
\green{<var>B - C * D</var>}<var>X</var> &amp;=&amp;
<var>C * E - A</var> \\ \\
\dfrac{\green{\cancel{<var>B - C * D</var>}}
<var>X</var>}{\green{\cancel{<var>B - C * D</var>}}}
&amp;=&amp; \dfrac{<var>C * E - A</var>}
{\green{<var>B - C * D</var>}} \\ \\
\end{eqnarray}
</code></p>
</div>
<div>
<p>Simplify:</p>
<p><code>\qquad
<var>X</var> = <var>fractionReduce(C * E - A, B - C * D)</var>
</code></p>
</div>
</div>
</div>
<div id="prob6"> <!-- Ax + B(C + Dx) = E -->
<div class="vars" data-ensure="(A + B * D !== 0) &&
(E - B * C !== 0) && (A + B * D !== 1)">
<var id="A">randRangeNonZero(-8, 8)</var>
<var id="B">
randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1])
</var>
<var id="C">randRangeNonZero(-10, 10)</var>
<var id="D">randRangeNonZero(-6, 6)</var>
<var id="E">randRangeNonZero(-10, 10)</var>
<var id="SOLUTION">(E - B * C) / (A + B * D)</var>
</div>
<p class="question">Solve for <code><var>X</var></code>:</p>
<p class="problem"><code>\qquad
<var>expr(["+", ["*", A, X], ["*", B, ["+", C, ["*", D, X]]]])</var> = <var>E</var>
</code></p>
<div class="solution" data-type="multiple">
<p>
<code><var>X</var> =</code>
<span class="sol"><var>SOLUTION</var></span>
</p>
</div>
<div class="hints">
<p>
Try simplifying the left side of the equation before
solving it.
</p>
<div>
<p data-if="B === -1">
Distribute the negative in front of the parentheses.
Be careful! The negative sign in front of the
parentheses means we're multiplying by
<code>\pink{-1}</code>:
</p><p data-else-if="B < 0">
Distribute the <code>\pink{<var>B</var>}</code>.
Be careful to pay attention to the negative sign
when you distribute:
</p><p data-else>
Distribute the <code>\pink{<var>B</var>}</code>:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["*", A, X])</var> +
\pink{<var>B</var>}\blue{(<var>expr(["+", C, ["*", D, X]])</var>)} &amp;=&amp;
<var>E</var> \\ \\
<var>expr(["*", A, X])</var> +
\pink{(<var>B</var>)}\blue{(<var>C</var>)} +
\pink{(<var>B</var>)}\blue{(<var>expr(["*", D, X])</var>)} &amp;=&amp;
<var>E</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>Multiply:</p>
<p><code>\qquad
<var>expr(["+", ["*", A, X], B * C, ["*", B * D, X]])</var> =
<var>E</var>
</code></p>
</div>
<div>
<p>
Combine the <span class="hint_blue">
<code><var>X</var></code> terms</span>:
</p>
<p><code>\qquad\begin{eqnarray}
\blue{<var>expr(["*", A, X])</var>} + <var>B * C</var> \blue{{}+ <var>expr(["*", B * D, X])</var>} &amp;=&amp; <var>E</var> \\ \\
\blue{<var>expr(["*", A + B * D, X])</var>} + <var>B * C</var> &amp;=&amp; <var>E</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
<var>B * C &lt; 0 ? "Add" : "Subtract"</var>
<code>\green{<var>abs(B * C)</var>}</code>
<var>B * C &lt; 0 ? "to" : "from"</var> both sides
to isolate the <code><var>X</var></code> term on the
left side:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["*", A + B * D, X])</var> \green{{} + <var>B * C</var>} &amp;=&amp; <var>E</var> \\ \\
\green{{}+<var>-B * C</var>} &amp;&amp;
\green{{}+<var>-B * C</var>} \\ \\
<var>expr(["*", A + B * D, X])</var> &amp;=&amp; <var>E - B * C</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Divide both sides by
<code>\green{<var>A + B * D</var>}</code>
to isolate <code><var>X</var></code>:
</p>
<p><code>\qquad\begin{eqnarray}
\green{<var>A + B * D</var>}<var>X</var> &amp;=&amp;
<var>E - B * C</var> \\ \\
\dfrac{\green{\cancel{<var>A + B * D</var>}}
<var>X</var>}{\green{\cancel{<var>A + B * D</var>}}}
&amp;=&amp; \dfrac{<var>E - B * C</var>}
{\green{<var>A + B * D</var>}} \\ \\
\end{eqnarray}
</code></p>
</div>
<div>
<p>Simplify:</p>
<p><code>\qquad
<var>X</var> = <var>fractionReduce(E - B * C, A + B * D)</var>
</code></p>
</div>
</div>
</div>
<div id="prob7"> <!-- A(Bx + C) = Dx + E -->
<div class="vars" data-ensure="(A * B - D !== 0) && (A * B - D !== 1)">
<var id="A">
randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1])
</var>
<var id="B">randRangeNonZero(-10, 10)</var>
<var id="C">randRange(1, 3)</var>
<var id="D">randRangeNonZero(-10, 10)</var>
<var id="E">randRangeNonZero(-10, 10)</var>
<var id="SOLUTION">(E - A * C) / (A * B - D)</var>
</div>
<p class="question">Solve for <code><var>X</var></code>:</p>
<p class="problem"><code>\qquad
<var>expr(["*", A, ["+", ["*", B, X], C]])</var> =
<var>expr(["+", ["*", D, X], E])</var>
</code></p>
<div class="solution" data-type="multiple">
<p>
<code><var>X</var> =</code>
<span class="sol"><var>SOLUTION</var></span>
</p>
</div>
<div class="hints">
<p>
Try simplifying the left side of the equation before
solving it.
</p>
<div>
<p data-if="A === -1">
Distribute the negative in front of the parentheses.
Be careful! The negative sign in front of the
parentheses means we're multiplying by
<code>\pink{-1}</code>:
</p><p data-else-if="A < 0">
Distribute the <code>\pink{<var>A</var>}</code>.
Be careful to pay attention to the negative sign
when you distribute:
</p><p data-else>
Distribute the <code>\pink{<var>A</var>}</code>:
</p>
<p><code>\qquad\begin{eqnarray}
\pink{<var>A</var>}\blue{(<var>expr(["+", ["*", B, X], C])</var>)} &amp;=&amp;
<var>expr(["+", ["*", D, X], E])</var> \\ \\
\pink{(<var>A</var>)}\blue{(<var>expr(["*", B, X])</var>)} +
\pink{(<var>A</var>)}\blue{(<var>C</var>)} &amp;=&amp;
<var>expr(["+", ["*", D, X], E])</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>Multiply:</p>
<p><code>\qquad
<var>expr(["+", ["*", A * B, X], A * C])</var> =
<var>expr(["+", ["*", D, X], E])</var>
</code></p>
</div>
<div>
<p>
<var>D &lt; 0 ? "Add" : "Subtract"</var>
<code>\green{<var>expr(["*", abs(D), X])</var>}
</code> <var>D &lt; 0 ? "to" : "from"</var> both
sides to eliminate the <code><var>X</var></code> term
from the right side:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["+", ["*", A * B, X], A * C])</var> &amp;=&amp;
\green{<var>expr(["*", D, X])</var>} + <var>E</var> \\ \\
\green{{}+<var>expr(["*", -D, X])</var>}
&amp;&amp; \green{{} +
<var>expr(["*", -D, X])</var>} \\ \\
<var>expr(["+", ["*", A * B, X], A * C])</var>
\green{{} + <var>expr(["*", -D, X])</var>} &amp;=&amp;
<var>E</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Combine the <span class="hint_blue">
<code><var>X</var></code> terms</span>:
</p>
<p><code>\qquad\begin{eqnarray}
\blue{<var>expr(["*", A * B, X])</var>} + <var>A * C</var>
\blue{{}+ <var>expr(["*", -D, X])</var>}
&amp;=&amp; <var>E</var> \\ \\
\blue{<var>expr(["*", A * B - D, X])</var>} + <var>A * C</var> &amp;=&amp; <var>E</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
<var>A * C &lt; 0 ? "Add" : "Subtract"</var>
<code>\green{<var>abs(A * C)</var>}</code>
<var>A * C &lt; 0 ? "to" : "from"</var> both sides
to isolate the <code><var>X</var></code> term on the
left side:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["*", A * B - D, X])</var> \green{{} + <var>A * C</var>} &amp;=&amp; <var>E</var> \\ \\
\green{{}+<var>-A * C</var>} &amp;&amp;
\green{{}+<var>-A * C</var>} \\ \\
<var>expr(["*", A * B - D, X])</var> &amp;=&amp; <var>E - A * C</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Divide both sides by
<code>\green{<var>A * B - D</var>}</code>
to isolate <code><var>X</var></code>:
</p>
<p><code>\qquad\begin{eqnarray}
\green{<var>A * B - D</var>}<var>X</var> &amp;=&amp;
<var>E - A * C</var> \\ \\
\dfrac{\green{\cancel{<var>A * B - D</var>}}
<var>X</var>}{\green{\cancel{<var>A * B - D</var>}}}
&amp;=&amp; \dfrac{<var>E - A * C</var>}
{\green{<var>A * B - D</var>}} \\ \\
\end{eqnarray}
</code></p>
</div>
<div>
<p>Simplify:</p>
<p><code>\qquad
<var>X</var> = <var>fractionReduce(E - A * C, A * B - D)</var>
</code></p>
</div>
</div>
</div>
<div id="prob8"> <!-- A(B + Cx) + D = E + Fx -->
<div class="vars" data-ensure="(A * B - F !== 0) &&
(A * C + D !== 0) && (E - A * C - D !== 0) && (A * B - F !== 1)">
<var id="A">
randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1])
</var>
<var id="B">randRangeNonZero(-10, 10)</var>
<var id="C">randRange(1, 3)</var>
<var id="D">randRangeNonZero(-10, 10)</var>
<var id="E">randRangeNonZero(-10, 10)</var>
<var id="F">randRangeNonZero(-10, 10)</var>
<var id="SOLUTION">(E - A * C - D) / (A * B - F)</var>
</div>
<p class="question">Solve for <code><var>X</var></code>:</p>
<p class="problem"><code>\qquad
<var>expr(["+", ["*", A, ["+", ["*", B, X], C]], D])</var> =
<var>expr(["+", E, ["*", F, X]])</var>
</code></p>
<div class="solution" data-type="multiple">
<p>
<code><var>X</var> =</code>
<span class="sol"><var>SOLUTION</var></span>
</p>
</div>
<div class="hints">
<p>
Try simplifying the left side of the equation before
solving it.
</p>
<div>
<p data-if="A === -1">
Distribute the negative in front of the parentheses.
Be careful! The negative sign in front of the
parentheses means we're multiplying by
<code>\pink{-1}</code>:
</p><p data-else-if="A < 0">
Distribute the <code>\pink{<var>A</var>}</code>.
Be careful to pay attention to the negative sign
when you distribute:
</p><p data-else>
Distribute the <code>\pink{<var>A</var>}</code>:
</p>
<p><code>\qquad\begin{eqnarray}
\pink{<var>A</var>}\blue{(<var>expr(["+", ["*", B, X], C])</var>)} + <var>D</var> &amp;=&amp;
<var>expr(["+", E, ["*", F, X]])</var> \\ \\
\pink{(<var>A</var>)}\blue{(<var>expr(["*", B, X])</var>)} +
\pink{(<var>A</var>)}\blue{(<var>C</var>)} + <var>D</var> &amp;=&amp;
<var>expr(["+", E, ["*", F, X]])</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>Multiply:</p>
<p><code>\qquad
<var>expr(["+", ["*", A * B, X], A * C, D])</var> =
<var>expr(["+", E, ["*", F, X]])</var>
</code></p>
</div>
<div>
<p>
<var>F &lt; 0 ? "Add" : "Subtract"</var>
<code>\green{<var>expr(["*", abs(F), X])</var>}
</code> <var>F &lt; 0 ? "to" : "from"</var> both
sides to eliminate the <code><var>X</var></code> term
from the right side:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["+", ["*", A * B, X], A * C, D])</var> &amp;=&amp;
<var>E</var> \green{{} + <var>expr(["*", F, X])</var>} \\ \\
\green{{}+<var>expr(["*", -F, X])</var>}
&amp;&amp; \green{{} +
<var>expr(["*", -F, X])</var>} \\ \\
<var>expr(["+", ["*", A * B, X], A * C, D])</var>
\green{{}+<var>expr(["*", -F, X])</var>} &amp;=&amp;
<var>E</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Combine the <span class="hint_blue">
<code><var>X</var></code> terms</span>:
</p>
<p><code>\qquad\begin{eqnarray}
\blue{<var>expr(["*", A * B, X])</var>} +
<var>expr(["+", A * C, D])</var>
\blue{{} + <var>expr(["*", -F, X])</var>} &amp;=&amp; <var>E</var> \\ \\
<var>expr(["+", A * C, D])</var>
\blue{{} + <var>expr(["*", A * B - F, X])</var>} &amp;=&amp; <var>E</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Combine the <span class="hint_blue"> numeric terms</span>:
</p>
<p><code>\qquad\begin{eqnarray}
\blue{<var>expr(["+", A * C, D])</var>} +
<var>expr(["*", A * B - F, X])</var> &amp;=&amp; <var>E</var> \\ \\
\blue{<var>A * C + D</var>} +
<var>expr(["*", A * B - F, X])</var> &amp;=&amp; <var>E</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
<var>A * C + D &lt; 0 ? "Add" : "Subtract"</var>
<code>\green{<var>abs(A * C + D)</var>}</code>
<var>A * C + D &lt; 0 ? "to" : "from"</var> both sides
to isolate the <code><var>X</var></code> term on the
left side:
</p>
<p><code>\qquad\begin{eqnarray}
\green{<var>A * C + D</var>} + <var>expr(["*", A * B - F, X])</var> &amp;=&amp; <var>E</var> \\ \\
\green{{}+<var>-(A * C + D)</var>} &amp;&amp;
\green{{}+<var>-(A * C + D)</var>} \\ \\
<var>expr(["*", A * B - F, X])</var> &amp;=&amp; <var>E - A * C - D</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Divide both sides by
<code>\green{<var>A * B - F</var>}</code>
to isolate <code><var>X</var></code>:
</p>
<p><code>\qquad\begin{eqnarray}
\green{<var>A * B - F</var>}<var>X</var> &amp;=&amp;
<var>E - A * C - D</var> \\ \\
\dfrac{\green{\cancel{<var>A * B - F</var>}}
<var>X</var>}{\green{\cancel{<var>A * B - F</var>}}}
&amp;=&amp; \dfrac{<var>E - A * C - D</var>}
{\green{<var>A * B - F</var>}} \\ \\
\end{eqnarray}
</code></p>
</div>
<div>
<p>Simplify:</p>
<p><code>\qquad
<var>X</var> = <var>fractionReduce(E - A * C - D, A * B - F)</var>
</code></p>
</div>
</div>
</div>
<div id="prob9"> <!-- A(Bx + C) = D(E + Fx) -->
<div class="vars" data-ensure="(A * B - D * F !== 0) && (A * B - D * F !== 1)">
<var id="A">
randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1])
</var>
<var id="B">randRangeNonZero(-10, 10)</var>
<var id="C">randRange(1, 3)</var>
<var id="D">
randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1])
</var>
<var id="E">randRangeNonZero(-10, 10)</var>
<var id="F">randRangeNonZero(-10, 10)</var>
<var id="SOLUTION">(D * E - A * C) / (A * B - D * F)</var>
</div>
<p class="question">Solve for <code><var>X</var></code>:</p>
<p class="problem"><code>\qquad
<var>expr(["*", A, ["+", ["*", B, X], C]])</var> =
<var>expr(["*", D, ["+", E, ["*", F, X]]])</var>
</code></p>
<div class="solution" data-type="multiple">
<p>
<code><var>X</var> =</code>
<span class="sol"><var>SOLUTION</var></span>
</p>
</div>
<div class="hints">
<p>
Try simplifying each side of the equation before
solving it.
</p>
<div>
<p data-if="A === -1">
Distribute the negative in front of the parentheses.
Be careful! The negative sign in front of the
parentheses means we're multiplying by
<code>\pink{-1}</code>:
</p><p data-else-if="A < 0">
Distribute the <code>\pink{<var>A</var>}</code>.
Be careful to pay attention to the negative sign
when you distribute:
</p><p data-else>
Distribute the <code>\pink{<var>A</var>}</code>:
</p>
<p><code>\qquad\begin{eqnarray}
\pink{<var>A</var>}\blue{(<var>expr(["+", ["*", B, X], C])</var>)} &amp;=&amp;
<var>expr(["*", D, ["+", E, ["*", F, X]]])</var> \\ \\
\pink{(<var>A</var>)}\blue{(<var>expr(["*", B, X])</var>)} +
\pink{(<var>A</var>)}\blue{(<var>C</var>)} &amp;=&amp;
<var>expr(["*", D, ["+", E, ["*", F, X]]])</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>Multiply:</p>
<p><code>\qquad
<var>expr(["+", ["*", A * B, X], A * C])</var> =
<var>expr(["*", D, ["+", E, ["*", F, X]]])</var>
</code></p>
</div>
<div>
<p data-if="D === -1">
Distribute the negative in front of the parentheses.
Be careful! The negative sign in front of the
parentheses means we're multiplying by
<code>\pink{-1}</code>:
</p><p data-else-if="D < 0">
Distribute the <code>\pink{<var>D</var>}</code>.
Be careful to pay attention to the negative sign
when you distribute:
</p><p data-else>
Distribute the <code>\pink{<var>D</var>}</code>:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["+", ["*", A * B, X], A * C])</var> &amp;=&amp;
\pink{(<var>D</var>)}\blue{(<var>expr(["+", E, ["*", F, X]])</var>)} \\ \\
<var>expr(["+", ["*", A * B, X], A * C])</var> &amp;=&amp;
\pink{(<var>D</var>)} \blue{(<var>E</var>)} +
\pink{(<var>D</var>)}
\blue{(<var>expr(["*", F, X])</var>)}
\end{eqnarray}
</code></p>
</div>
<div>
<p>Multiply:</p>
<p><code>\qquad
<var>expr(["+", ["*", A * B, X], A * C])</var> =
<var>expr(["+", D * E, ["*", D * F, X]])</var>
</code></p>
</div>
<div>
<p>
<var>D * F &lt; 0 ? "Add" : "Subtract"</var>
<code>\green{<var>expr(["*", abs(D * F), X])</var>}
</code> <var>D * F &lt; 0 ? "to" : "from"</var> both
sides to eliminate the <code><var>X</var></code> term
from the right side:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["+", ["*", A * B, X], A * C])</var> &amp;=&amp;
<var>D * E</var> \green{{} + <var>expr(["*", D * F, X])</var>} \\ \\
\green{{}+<var>expr(["*", -(D * F), X])</var>}
&amp;&amp; \green{{} +
<var>expr(["*", -(D * F), X])</var>} \\ \\
<var>expr(["+", ["*", A * B, X], A * C])</var>
\green{+<var>expr(["*", -(D * F), X])</var>} &amp;=&amp;
<var>D * E</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Combine the <span class="hint_blue">
<code><var>X</var></code> terms</span>:
</p>
<p><code>\qquad\begin{eqnarray}
\blue{<var>expr(["*", A * B, X])</var>} + <var>A * C</var>
\blue{+<var>expr(["*", -(D * F), X])</var>} &amp;=&amp;
<var>D * E</var> \\ \\
\blue{<var>expr(["*", A * B - D * F, X])</var>} + <var>A * C</var> &amp;=&amp;
<var>D * E</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
<var>A * C &lt; 0 ? "Add" : "Subtract"</var>
<code>\green{<var>abs(A * C)</var>}</code>
<var>A * C &lt; 0 ? "to" : "from"</var> both sides
to isolate the <code><var>X</var></code> term on the
left side:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["*", A * B - D * F, X])</var> + \green{<var>A * C</var>} &amp;=&amp;
<var>D * E</var> \\ \\
\green{{}+<var>-A * C</var>} &amp;&amp;
\green{{}+<var>-A * C</var>} \\ \\
<var>expr(["*", A * B - D * F, X])</var>
&amp;=&amp; <var>D * E - A * C</var>
\end{eqnarray}
</code></p>
</div>
<div>
<p>
Divide both sides by
<code>\green{<var>A * B - D * F</var>}</code>
to isolate <code><var>X</var></code>:
</p>
<p><code>\qquad\begin{eqnarray}
<var>expr(["*", A * B - D * F, X])</var>
&amp;=&amp; <var>D * E - A * C</var> \\ \\
\dfrac{\green{\cancel{<var>A * B - D * F</var>}}
<var>X</var>}{\green{\cancel{<var>A * B - D * F</var>}}}
&amp;=&amp; \dfrac{<var>D * E - A * C</var>}
{\green{<var>A * B - D * F</var>}} \\ \\
\end{eqnarray}
</code></p>
</div>
<div>
<p>Simplify:</p>
<p><code>\qquad
<var>X</var>
= <var>fractionReduce(D * E - A * C, A * B - D * F)</var>
</code></p>
</div>
</div>
</div>
</div>
</div>
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