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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>Systems of equations with elimination</title>
<script src="../khan-exercise.js"></script>
<style type="text/css">
#answer_area input[type=text] {
width: 50px;
}
</style>
</head>
<body>
<div class="exercise">
<div class="vars">
<var id="X">randRange( 1, 10 )</var>
<var id="Y" data-ensure="X !== Y">randRange( 1, 10 )</var>
<var id="INDEX">0</var>
<var id="Z1">randRangeWeighted( 1, 3, 1, 0.5 ) * randRangeNonZero( -1, 1 )</var>
<var id="Z2" data-ensure="abs( Z2 ) <= 6 && Z2 !== 0">INDEX === 0 ? -1 * Z1 : randRangeWeighted( -4, 4, -1, 0 ) * Z1</var>
<var id="Z3">randRange( 2, 6 ) * randRangeNonZero( -1, 1 )</var>
<var id="Z4" data-ensure="getGCD( abs( Z3 ), abs( Z4 ) ) === 1">randRange( 2, 6 ) * randRangeNonZero( -1, 1 )</var>
<var id="Z5">randRange( 2, 6 ) * randRangeNonZero( -1, 1 )</var>
<var id="Z6" data-ensure="getGCD( abs( Z5 ), abs( Z6 ) ) === 1 && ( Z3 * Z6 ) !== ( Z4 * Z5 )">randRange( 2, 6 ) * randRangeNonZero( -1, 1 )</var>
</div>
<div class="problems">
<div id="elimination-x">
<div class="vars">
<var id="A1">[ Z1, Z1, Z2, Z5 ][ INDEX ]</var>
<var id="B1">Z3</var>
<var id="C1">A1 * X + B1 * Y</var>
<var id="A2">[ Z2, Z2, Z1, Z6 ][ INDEX ]</var>
<var id="B2">Z4</var>
<var id="C2">A2 * X + B2 * Y</var>
<var id="MULT1">[ 1, -A2 / A1, 1, A2 &gt; 0 &amp;&amp; A1 &lt; 0 ? A2 : -A2 ][ INDEX ]</var>
<var id="MULT2">[ 1, 1, -A1 / A2, A2 &gt; 0 &amp;&amp; A1 &lt; 0 ? -A1 : A1 ][ INDEX ]</var>
<var id="SIGN_1">B1 * Y &gt; 0 ? "-" : "+"</var>
<var id="SIGN_2">B2 * Y &gt; 0 ? "-" : "+"</var>
</div>
<p class="problem">Solve for <code>x</code> and <code>y</code> using elimination.</p>
<div class="question">
<p><code>\color{<var>BLUE</var>}{<var>expr(["+", ["*", A1, "x"], ["*", B1, "y"]])</var> = <var>C1</var>}</code><br />
<code class="hint_green"><var>expr(["+", ["*", A2, "x"], ["*", B2, "y"]])</var> = <var>C2</var></code></p>
</div>
<div class="solution" data-type="multiple">
<p><code>x</code> = <span class="sol"><var>X</var></span></p>
<p><code>y</code> = <span class="sol"><var>Y</var></span></p>
</div>
<div class="hints">
<p>We can eliminate <code>x</code> by adding the equations together when the <code>x</code> coefficients have opposite signs.</p>
<div>
<p>Add the equations together. Notice that the terms <code class="hint_blue"><var>expr(["*", A1, "x"])</var></code> and <code class="hint_green"><var>expr(["*", A2, "x"])</var></code> cancel out.</p>
</div>
<div>
<p data-if="( B1 * MULT1 + B2 * MULT2 ) !== 1"><code><var>expr(["*", B1 * MULT1 + B2 * MULT2, "y"])</var> = <var>C1 * MULT1 + C2 * MULT2</var></code></p>
<p data-if="( B1 * MULT1 + B2 * MULT2 ) !== 1"><code>\dfrac{<var>expr(["*", B1 * MULT1 + B2 * MULT2, "y"])</var>}{\color{<var>BLUE</var>}{<var>B1 * MULT1 + B2 * MULT2</var>}} = \dfrac{<var>C1 * MULT1 + C2 * MULT2</var>}{\color{<var>BLUE</var>}{<var>B1 * MULT1 + B2 * MULT2</var>}}</code></p>
<p><code>\color{<var>ORANGE</var>}{y = <var>Y</var>}</code></p>
</div>
<div>
<p>Now that you know <code>\color{<var>ORANGE</var>}{y = <var>Y</var>}</code>, plug it back into <code>\thinspace \color{<var>BLUE</var>}{<var>expr(["+", ["*", A1, "x"], ["*", B1, "y"]])</var> = <var>C1</var>}\thinspace</code> to find <code>x</code>.</p>
</div>
<div>
<p><code>\color{<var>BLUE</var>}{<var>expr(["*", A1, "x"])</var> + <span data-if="abs( B1 ) !== 1"><var>B1</var></span><span data-if="B1 === -1">-</span>}\color{<var>ORANGE</var>}{(<var>Y</var>)}\color{<var>BLUE</var>}{= <var>C1</var>}</code></p>
</div>
<div>
<p data-if="B1 !== 1"><code><var>expr(["+", ["*", A1, "x"], B1 * Y])</var> = <var>C1</var></code></p>
<p><code><var>expr(["+", ["*", A1, "x"], B1 * Y])</var>\color{<var>BLUE</var>}{<var>SIGN_1</var><var>abs( B1 * Y )</var>} = <var>C1</var>\color{<var>BLUE</var>}{<var>SIGN_1</var><var>abs( B1 * Y )</var>}</code></p>
<p data-if="A1 !== 1"><code><var>expr(["*", A1, "x"])</var> = <var>C1 - B1 * Y</var></code></p>
<p data-if="A1 !== 1"><code>\dfrac{<var>expr(["*", A1, "x"])</var>}{\color{<var>BLUE</var>}{<var>A1</var>}} = \dfrac{<var>C1 - B1 * Y</var>}{\color{<var>BLUE</var>}{<var>A1</var>}}</code></p>
<p><code>\color{red}{x = <var>X</var>}</code></p>
</div>
<div>
<p>You can also plug <code>\color{<var>ORANGE</var>}{y = <var>Y</var>}</code> into <code>\thinspace \color{<var>GREEN</var>}{<var>expr(["+", ["*", A2, "x"], ["*", B2, "y"]])</var> = <var>C2</var>}\thinspace</code> and get the same answer for <code>x</code>:</p>
<p><code>\color{<var>GREEN</var>}{<var>expr(["*", A2, "x"])</var> + <span data-if="abs( B2 ) !== 1"><var>B2</var></span><span data-if="B2 === -1">-</span>}\color{<var>ORANGE</var>}{(<var>Y</var>)}\color{<var>GREEN</var>}{= <var>C2</var>}</code></p>
<p><code>\color{red}{x = <var>X</var>}</code></p>
</div>
</div>
</div>
<div id="elimination-y">
<div class="vars">
<var id="A1">Z3</var>
<var id="B1">[ Z1, Z1, Z2, Z5 ][ INDEX ]</var>
<var id="C1">A1 * X + B1 * Y</var>
<var id="A2">Z4</var>
<var id="B2">[ Z2, Z2, Z1, Z6 ][ INDEX ]</var>
<var id="C2">A2 * X + B2 * Y</var>
<var id="MULT1">[ 1, -B2 / B1, 1, B2 &gt; 0 &amp;&amp; B1 &lt; 0 ? B2 : -B2 ][ INDEX ]</var>
<var id="MULT2">[ 1, 1, -B1 / B2, B2 &gt; 0 &amp;&amp; B1 &lt; 0 ? -B1 : B1 ][ INDEX ]</var>
<var id="SIGN_1">A1 * X &gt; 0 ? "-" : "+"</var>
<var id="SIGN_2">A2 * X &gt; 0 ? "-" : "+"</var>
</div>
<p class="problem">Solve for <code>x</code> and <code>y</code> using elimination.</p>
<div class="question">
<p><code>\color{<var>BLUE</var>}{<var>expr(["+", ["*", A1, "x"], ["*", B1, "y"]])</var> = <var>C1</var>}</code><br />
<code class="hint_green"><var>expr(["+", ["*", A2, "x"], ["*", B2, "y"]])</var> = <var>C2</var></code></p>
</div>
<div class="solution" data-type="multiple">
<p><code>x</code> = <span class="sol"><var>X</var></span></p>
<p><code>y</code> = <span class="sol"><var>Y</var></span></p>
</div>
<div class="hints">
<p>We can eliminate <code>y</code> by adding the equations together when the <code>y</code> coefficients have opposite signs.</p>
<div>
<p>Add the equations together. Notice that the terms <code class="hint_blue"><var>expr(["*", B1, "y"])</var></code> and <code class="hint_green"><var>expr(["*", B2, "y"])</var></code> cancel out.</p>
</div>
<div>
<p data-if="( A1 * MULT1 + A2 * MULT2 ) !== 1"><code><var>expr(["*", A1 * MULT1 + A2 * MULT2, "x"])</var> = <var>C1 * MULT1 + C2 * MULT2</var></code></p>
<p data-if="( A1 * MULT1 + A2 * MULT2 ) !== 1"><code>\dfrac{<var>expr(["*", A1 * MULT1 + A2 * MULT2, "x"])</var>}{\color{<var>BLUE</var>}{<var>A1 * MULT1 + A2 * MULT2</var>}} = \dfrac{<var>C1 * MULT1 + C2 * MULT2</var>}{\color{<var>BLUE</var>}{<var>A1 * MULT1 + A2 * MULT2</var>}}</code></p>
<p><code>\color{red}{x = <var>X</var>}</code></p>
</div>
<div>
<p>Now that you know <code>\color{red}{x = <var>X</var>}</code>, plug it back into <code>\thinspace \color{<var>BLUE</var>}{<var>expr(["+", ["*", A1, "x"], ["*", B1, "y"]])</var> = <var>C1</var>}\thinspace</code> to find <code>y</code>.</p>
</div>
<div>
<p><code>\color{<var>BLUE</var>}{<span data-if="abs( A1 ) !== 1"><var>A1</var></span><span data-if="A1 === -1">-</span>}\color{red}{(<var>X</var>)}\color{<var>BLUE</var>}{ + <var>expr(["*", B1, "y"])</var> = <var>C1</var>}</code></p>
</div>
<div>
<p data-if="A1 !== 1"><code><var>expr(["+", A1 * X, ["*", B1, "y"]])</var> = <var>C1</var></code></p>
<p><code><var>A1 * X</var>\color{<var>BLUE</var>}{<var>SIGN_1</var><var>abs( A1 * X )</var>} + <var>expr(["*", B1, "y"])</var> = <var>C1</var>\color{<var>BLUE</var>}{<var>SIGN_1</var><var>abs( A1 * X )</var>}</code></p>
<p data-if="B1 !== 1"><code><var>expr(["*", B1, "y"])</var> = <var>C1 - A1 * X</var></code></p>
<p data-if="B1 !== 1"><code>\dfrac{<var>expr(["*", B1, "y"])</var>}{\color{<var>BLUE</var>}{<var>B1</var>}} = \dfrac{<var>C1 - A1 * X</var>}{\color{<var>BLUE</var>}{<var>B1</var>}}</code></p>
<p><code>\color{<var>ORANGE</var>}{y = <var>Y</var>}</code></p>
</div>
<div>
<p>You can also plug <code>\color{red}{x = <var>X</var>}</code> into <code>\thinspace \color{<var>GREEN</var>}{<var>expr(["+", ["*", A2, "x"], ["*", B2, "y"]])</var> = <var>C2</var>}\thinspace</code> and get the same answer for <code>y</code>:</p>
<p><code>\color{<var>GREEN</var>}{<span data-if="abs( A2 ) !== 1"><var>A2</var></span><span data-if="A2 === -1">-</span>}\color{red}{(<var>X</var>)}\color{<var>GREEN</var>}{ + <var>expr(["*", B2, "y"])</var> = <var>C2</var>}</code></p>
<p><code>\color{<var>ORANGE</var>}{y = <var>Y</var>}</code></p>
</div>
</div>
</div>
</div>
</div>
</body>
</html>
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