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solving_for_a_variable.html
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solving_for_a_variable.html
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<!DOCTYPE html>
<html data-require="math math-format">
<head>
<meta charset="UTF-8" />
<title>Solving for a variable</title>
<script src="../khan-exercise.js"></script>
<script type="text/javascript">
function color( str, blue ) {
var pre = "";
if ( typeof str === "number" && str < 0 ) {
pre = "-";
str = Math.abs( str );
} else if ( typeof str === "string" && str.indexOf( "-" ) === 0 ) {
pre = "-";
str = str.slice( 1 );
}
return pre + "\\color{" + ( blue ? "blue" : "green" ) + "}{" + str + "}";
}
</script>
</head>
<body>
<div class="exercise">
<div class="vars">
<var id="LETTERS">"abcdefghijklmnpqrstuvwxyz"</var>
<var id="LETTER">randRange( 0, LETTERS.length - 3 )</var>
<var id="X">LETTERS[ LETTER + 0 ]</var>
<var id="Y">LETTERS[ LETTER + 1 ]</var>
<var id="Z">LETTERS[ LETTER + 2 ]</var>
</div>
<div class="problems">
<!-- Gx(yz) + Ay + Bz + C = Dy + Ez + F -->
<!-- The (yz) means that a y or a z may both be included in that term. -->
<div id="GABC_DEF">
<div class="vars">
<!-- pick random integers for coefficients, making sure that they don't cancel each other out on each side of the equation -->
<var id="A">randRangeNonZero( -10, 10 )</var>
<var id="B">randRangeNonZero( -10, 10 )</var>
<var id="C">randRangeNonZero( -10, 10 )</var>
<var id="D" data-ensure="A !== D">randRangeNonZero( -10, 10 )</var>
<var id="E" data-ensure="B !== E">randRangeNonZero( -10, 10 )</var>
<var id="F" data-ensure="C !== F">randRangeNonZero( -10, 10 )</var>
<var id="G">randRangeNonZero( -10, 10 )</var>
<var id="X_HAS_Y">random() < 0.5</var>
<var id="X_HAS_Z">random() < 0.5</var>
<var id="X_TERM">G + X + ( X_HAS_Y ? Y : "" ) + ( X_HAS_Z ? Z : "" )</var>
<var id="X_EXTRAS">( X_HAS_Y ? Y : "" ) + ( X_HAS_Z ? Z : "" )</var>
<var id="GCD">getGCD( getGCD( abs( G ), abs( D-A ) ), getGCD( abs( E-B ), abs( F-C ) ) )</var>
<var id="DIVISOR">( G < 0 ? -1 : 1 ) / GCD</var>
<var id="ANSWER">function( y, z, c, d ) {
return X + "= \\dfrac{" + plus( round( DIVISOR*y )+Y, round( DIVISOR*z )+Z, round( DIVISOR*c ) ) + "}{" + plus( round( DIVISOR*d )+X_EXTRAS ) + "}";
}</var>
</div>
<p class="problem"><code><var>plus( X_TERM, A+Y, B+Z, C )</var> = <var>plus( D+Y, E+Z, F )</var></code></p>
<p class="question">Solve for <code><var>X</var></code>.</p>
<p class="solution"><code><var>ANSWER( D-A, E-B, F-C, G )</var></code></p>
<ul class="choices" data-show="4" data-none="true">
<li data-if="D+A !== 0"><code><var>ANSWER( D+A, E-B, F-C, G )</var></code></li>
<li data-if="E+B !== 0"><code><var>ANSWER( D-A, E+B, F-C, G )</var></code></li>
<li data-if="F+C !== 0"><code><var>ANSWER( D-A, E-B, F+C, G )</var></code></li>
<li data-if="D+A !== 0 && E+B !== 0"><code><var>ANSWER( D+A, E+B, F-C, G )</var></code></li>
<li data-if="D+A !== 0 && F+C !== 0"><code><var>ANSWER( D+A, E-B, F+C, G )</var></code></li>
<li data-if="E+B !== 0 && F+C !== 0"><code><var>ANSWER( D-A, E+B, F+C, G )</var></code></li>
<li data-if="E+B !== 0 && E+B !== 0 && F+C !== 0"><code><var>ANSWER( D+A, E+B, F+C, G )</var></code></li>
</ul>
<div class="hints">
<div>
<p>Combine constant terms on the right.</p>
<p><code><var>plus( X_TERM, A+Y, B+Z, color( C, true ) )</var> = <var>plus( D+Y, E+Z, color( F, true ) )</var></code></p>
</div>
<p><code><var>plus( X_TERM, A+Y, B+Z )</var> = <var>plus( D+Y, E+Z, color( F-C, true ) )</var></code></p>
<div>
<p>Combine <code><var>Z</var></code> terms on the right.</p>
<p><code><var>plus( X_TERM, A+Y, color( B+Z, false ) )</var> = <var>plus( D+Y, color( E+Z, false ), F-C )</var></code></p>
</div>
<p><code><var>plus( X_TERM, A+Y )</var> = <var>plus( D+Y, color( (E-B)+Z, false ), F-C )</var></code></p>
<div>
<p>Combine <code><var>Y</var></code> terms on the right.</p>
<p><code><var>plus( X_TERM, color( A+Y, true ) )</var> = <var>plus( color( D+Y, true ), (E-B)+Z, F-C )</var></code></p>
</div>
<p><code><var>plus( X_TERM )</var> = <var>plus( color( (D-A)+Y, true ), (E-B)+Z, F-C )</var></code></p>
<div>
<p>Isolate <code><var>X</var></code>.</p>
<p><code><var>plus( color( G, false ) + X + color( X_EXTRAS, false ) )</var> = <var>plus( (D-A)+Y, (E-B)+Z, F-C )</var></code></p>
</div>
<p><code><var>X</var> = \dfrac{ <var>plus( (D-A)+Y, (E-B)+Z, F-C )</var> }{ <var>plus( color( G + X_EXTRAS, false ) )</var> }</code></p>
<div data-if="GCD > 1 || G < 0" data-unwrap>
<p data-if="GCD > 1">All of these terms are divisible by <code><var>GCD</var></code>.</p>
<p data-if="GCD > 1 && G < 0">Divide by the common factor and swap signs so the denominator isn't negative.</p>
<p data-else-if="GCD > 1">Divide by the common factor.</p>
<p data-else>Swap the signs so the denominator isn't negative.</p>
<p><code><var>X</var> = \dfrac{ <var>plus( color( round( DIVISOR*(D-A) ), true )+Y, color( round( DIVISOR*(E-B) ), true )+Z, color( round( DIVISOR*(F-C) ), true ) )</var> }{ <var>plus( color( round( DIVISOR*G )+X_EXTRAS, true ) )</var> }</code></p>
</div>
</div>
</div>
<!-- Axy + Bxz + Cx + D = Ey + F -->
<div id="ABCD_EF">
<div class="vars">
<var id="A">randRangeNonZero( -10, 10 )</var>
<var id="B">randRangeNonZero( -10, 10 )</var>
<var id="C">randRangeNonZero( -10, 10 )</var>
<var id="D">randRangeNonZero( -10, 10 )</var>
<var id="E">randRangeNonZero( -10, 10 )</var>
<var id="F" data-ensure="D !== F">randRangeNonZero( -10, 10 )</var>
<var id="ANSWER">function( y1, c, y2, z, d, flip ) {
if ( typeof flip === "undefined" ? FLIP : flip ) {
y1 = -y1;
c = -c;
y2 = -y2;
z = -z;
d = -d;
}
return X + "= \\dfrac{" + plus( y1+Y, c ) + "}{" + plus( y2+Y, z+Z, d ) + "}";
}</var>
<var id="FLIP">(function() {
var n = 0,
ns = [ E, F-D, A, B, C ];
for( var i = 0; i < ns.length; i++ ) {
(ns[ i ] < 0 ) && n++;
}
return n >= 3;
})()</var>
</div>
<p class="problem"><code><var>plus( A+X+Y, B+X+Z, C+X, D )</var> = <var>plus( E+Y, F )</var></code></p>
<p class="question">Solve for <code><var>X</var></code>.</p>
<p class="solution"><code><var>ANSWER( E, F-D, A, B, C )</var></code></p>
<ul class="choices" data-show="4" data-none="true">
<li><code><var>ANSWER( E, F+D, A, B, C )</var></code></li>
<li><code><var>ANSWER( 0, F-D, A, B, C )</var></code></li>
<li><code><var>ANSWER( E, 0, A, B, C )</var></code></li>
<li><code><var>ANSWER( E, F-D, 0, B, C )</var></code></li>
<li><code><var>ANSWER( E, F-D, A, 0, C )</var></code></li>
<li><code><var>ANSWER( E, F-D, A, B, 0 )</var></code></li>
<li><code><var>ANSWER( E+A, F-D, A, B, C )</var></code></li>
<li><code><var>ANSWER( E-A, F-D, A, B, C )</var></code></li>
<li><code><var>ANSWER( E, F-D, A+B, B, C )</var></code></li>
<li><code><var>ANSWER( E, F-D, A-B, B, C )</var></code></li>
<li><code><var>ANSWER( E, F-D, A, A+B, C )</var></code></li>
<li><code><var>ANSWER( E, F-D, A, A-B, C )</var></code></li>
<li><code><var>ANSWER( E, F-D, A, B, A+C )</var></code></li>
<li><code><var>ANSWER( E, F-D, A, B, A-C )</var></code></li>
</ul>
<div class="hints">
<div>
<p>Combine constant terms on the right.</p>
<p><code><var>plus( A+X+Y, B+X+Z, C+X, color( D, true ) )</var> = <var>plus( E+Y, color( F, true ) )</var></code></p>
</div>
<p><code><var>plus( A+X+Y, B+X+Z, C+X )</var> = <var>plus( E+Y, color( F-D, true ) )</var></code></p>
<div>
<p>Notice that all the terms on the left-hand side of the equation have <code><var>X</var></code> in them.</p>
<p><code><var>plus( A+color( X, false )+Y, B+color( X, false )+Z, C+color( X, false ) )</var> = <var>plus( E+Y, F-D )</var></code></p>
</div>
<p>Factor out the <code><var>X</var></code>.</p>
<p><code><var>color( X, false )</var> \cdot \left( <var>plus( A+Y, B+Z, C )</var> \right) = <var>plus( E+Y, F-D )</var></code></p>
<div>
<p>Isolate the <code><var>X</var></code>.</p>
<p><code><var>X</var> \cdot \left( <var>color( plus( A+Y, B+Z, C ), true )</var> \right) = <var>plus( E+Y, F-D )</var></code></p>
</div>
<p><code><var>X</var> = \dfrac{ <var>plus( E+Y, F-D )</var> }{ <var>color( plus( A+Y, B+Z, C ), true )</var> }</code></p>
<div data-if="FLIP">
<p>We can simplify this by multiplying the top and bottom by <code>-1</code>.</p>
<p><code><var>ANSWER( E, F-D, A, B, C )</var></code></p>
</div>
</div>
</div>
<!-- (Ax + By) / C = (Dx + Ez) / F -->
<div id="ABC_DEF">
<div class="vars" data-ensure="(A_TERM-D_TERM)*DIVISOR !== 1 && (A_TERM-D_TERM)*DIVISOR !== -1 && (A_TERM+D_TERM)*DIVISOR !== 1 && (A_TERM+D_TERM)*DIVISOR !== -1">
<var id="A">randRangeNonZero( -10, 10 )</var>
<var id="B">randRangeNonZero( -10, 10 )</var>
<var id="C">randRangeExclude( -10, 10, [ 0, -1, 1 ] )</var>
<var id="D">randRangeNonZero( -10, 10 )</var>
<var id="E">randRangeNonZero( -10, 10 )</var>
<var id="F" data-ensure="A*F !== D*C">randRangeExclude( -10, 10, [ 0, -1, 1 ] )</var>
<var id="A_TERM">( function() {
if ( C === F || C === -F || C % F === 0 ) {
return A;
} else {
return A*F;
}
})()</var>
<var id="B_TERM">( function() {
if ( C === F || C === -F || C % F === 0 ) {
return B;
} else {
return B*F;
}
})()</var>
<var id="D_TERM">( function() {
if ( C === F ) {
return D;
} else if ( C === -F ) {
return -D;
} else if ( C % F === 0 ) {
return D*(C/F);
} else {
return D*C;
}
})()</var>
<var id="E_TERM">( function() {
if ( C === F ) {
return E;
} else if ( C === -F ) {
return -E;
} else if ( C % F === 0 ) {
return E*(C/F);
} else {
return E*C;
}
})()</var>
<var id="GCD">getGCD( getGCD( abs( E_TERM ), abs( B_TERM ) ), abs( A_TERM - D_TERM ) )</var>
<var id="DIVISOR">( A_TERM-D_TERM < 0 ? -1 : 1 ) / GCD</var>
<var id="ANSWER">function( z, y, c ) {
return X + " = \\dfrac{" + plus( round( z*DIVISOR )+Z, round( y*DIVISOR )+Y ) + "}{" + round( c*DIVISOR ) + "}";
}</var>
</div>
<p class="problem"><code>\dfrac{ <var>plus( A+X, B+Y )</var> }{ <var>C</var> } = \dfrac{ <var>plus( D+X, E+Z )</var> }{ <var>F</var> }</code></p>
<p class="question">Solve for <code><var>X</var></code>.</p>
<p class="solution"><code><var>ANSWER( E_TERM, -B_TERM, A_TERM-D_TERM )</var></code></p>
<ul class="choices" data-none="true" data-show="4">
<li><code><var>ANSWER( E_TERM, B_TERM, A_TERM-D_TERM )</var></code></li>
<li><code><var>ANSWER( -E_TERM, -B_TERM, A_TERM-D_TERM )</var></code></li>
<li><code><var>ANSWER( -E_TERM, B_TERM, A_TERM-D_TERM )</var></code></li>
<li data-if="A_TERM+D_TERM !== 0" data-unwrap><ul data-unwrap>
<li><code><var>ANSWER( E_TERM, -B_TERM, A_TERM+D_TERM )</var></code></li>
<li><code><var>ANSWER( E_TERM, B_TERM, A_TERM+D_TERM )</var></code></li>
<li><code><var>ANSWER( -E_TERM, -B_TERM, A_TERM+D_TERM )</var></code></li>
<li><code><var>ANSWER( -E_TERM, B_TERM, A_TERM+D_TERM )</var></code></li>
</ul></li>
<li data-if="-A_TERM-D_TERM !== 0" data-unwrap><ul data-unwrap>
<li><code><var>ANSWER( E_TERM, -B_TERM, -A_TERM-D_TERM )</var></code></li>
<li><code><var>ANSWER( E_TERM, B_TERM, -A_TERM-D_TERM )</var></code></li>
<li><code><var>ANSWER( -E_TERM, B_TERM, -A_TERM-D_TERM )</var></code></li>
<li><code><var>ANSWER( -E_TERM, -B_TERM, -A_TERM-D_TERM )</var></code></li>
</ul></li>
</ul>
<div class="hints">
<div data-if="C === F || C === -F" data-unwrap>
<div>
<p>Notice that the left- and right- denominators are <span data-if="C === F">the same</span><span data-else>opposite</span>.</p>
<p><code>\dfrac{ <var>plus( A+X, B+Y )</var> }{ <var>color( C, true )</var> } = \dfrac{ <var>plus( D+X, E+Z )</var> }{ <var>color( F, true )</var> }</code></p>
</div>
<div>
<p>So we can multiply both sides by <code><var>C</var></code>.</p>
<p><code><var>color( C, true )</var> \cdot \dfrac{ <var>plus( A+X, B+Y )</var> }{ <var>color( C, true )</var> } = <var>color( C, true )</var> \cdot \dfrac{ <var>plus( D+X, E+Z )</var> }{ <var>color( F, true )</var> }</code></p>
</div>
<p><code><var>plus( A+X, B+Y )</var> =
<span data-if="C === F"><var>plus( D+X, E+Z )</var></span>
<span data-else><var>color( "-", true )</var> \cdot \left( <var>plus( D+X, E+Z )</var> \right)</span>
</code></p>
<div data-if="C === -F" data-unwrap>
<div>
<p>Distribute the negative sign on the right side.</p>
<p><code><var>plus( A+X, B+Y )</var> = <var>plus( D_TERM+X, E_TERM+Z )</var></code></p>
</div>
<p><code><var>plus( color( A_TERM, true )+X, color( B_TERM, true )+Y )</var> = <var>plus( color( D_TERM, true )+X, color( E_TERM, true )+Z )</var></code></p>
</div>
</div>
<div data-else data-unwrap>
<div>
<p>Multiply both sides by the left denominator.</p>
<p><code>\dfrac{ <var>plus( A+X, B+Y )</var> }{ <var>color( C, true )</var> } = \dfrac{ <var>plus( D+X, E+Z )</var> }{ <var>F</var> }</code></p>
</div>
<p><code><var>color( C, true )</var> \cdot \dfrac{ <var>plus( A+X, B+Y )</var> }{ <var>color( C, true )</var> } = <var>color( C, true )</var> \cdot \dfrac{ <var>plus( D+X, E+Z )</var> }{ <var>F</var> }</code></p>
<p><code><var>plus( A+X, B+Y )</var> = <var>color( C, true )</var> \cdot \dfrac { <var>plus( D+X, E+Z )</var> }{ <var>F</var> }</code></p>
<div data-if="C % F === 0" data-unwrap>
<div>
<p>Reduce the right side.</p>
<p><code><var>plus( A+X, B+Y )</var> = <var>color( C, false )</var> \cdot \dfrac{ <var>plus( D+X, E+Z )</var> }{ <var>color( F, false )</var> }</code></p>
</div>
<p><code><var>plus( A+X, B+Y )</var> = <var>color( C / F, false )</var> \cdot \left( <var>plus( D+X, E+Z )</var> \right)</code></p>
</div><div data-else>
<div>
<p>Multiply both sides by the right denominator.</p>
<p><code><var>plus( A+X, B+Y )</var> = <var>C</var> \cdot \dfrac{ <var>plus( D+X, E+Z )</var> }{ <var>color( F, false )</var> }</code></p>
</div>
<p><code><var>color( F, false )</var> \cdot \left( <var>plus( A+X, B+Y )</var> \right) = <var>color( F, false )</var> \cdot <var>C</var> \cdot \dfrac{ <var>plus( D+X, E+Z )</var> }{ <var>color( F, false )</var> }</code></p>
<p><code><var>color( F, false )</var> \cdot \left( <var>plus( A+X, B+Y )</var> \right) = <var>C</var> \cdot \left( <var>plus( D+X, E+Z )</var> \right)</code></p>
</div>
<div>
<p>Distribute <span data-if="C % F === 0">the right side</span><span data-else>both sides</span>.</p>
<p data-if="C % F === 0"><code><var>plus( A+X, B+Y )</var> = <var>color( C / F, true )</var> \cdot \left( <var>plus( color( D+X, true ), color( E+Z, true ) )</var> \right)</code></p>
<p data-else><code><var>color( F, true )</var> \cdot \left( <var>plus( A+X, B+Y )</var> \right) = <var>color( C, true )</var> \cdot \left( <var>plus( D+X, E+Z )</var> \right)</code></p>
</div>
<p data-if="C % F === 0"><code><var>plus( A_TERM+X, B_TERM+Y )</var> = <var>plus( color( D_TERM, true )+X, color( E_TERM, true )+Z )</var></code></p>
<p data-else><code><var>plus( color( A_TERM, true )+X, color( B_TERM, true )+Y )</var> = <var>plus( color( D_TERM, true )+X, color( E_TERM, true )+Z )</var></code></p>
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<p>Combine <code><var>X</var></code> terms on the left.</p>
<p><code><var>plus( color( A_TERM+X, false ), B_TERM+Y )</var> = <var>plus( color( D_TERM+X, false ), E_TERM+Z )</var></code></p>
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<p><code><var>plus( color( (A_TERM-D_TERM)+X, false ), (B_TERM)+Y )</var> = <var>(E_TERM)+Z</var></code></p>
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<p>Move the <code><var>Y</var></code> term to the right.</p>
<p><code><var>plus( (A_TERM-D_TERM)+X, color( B_TERM+Y, true ) )</var> = <var>E_TERM+Z</var></code></p>
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<p><code><var>(A_TERM-D_TERM)+X</var> = <var>plus( E_TERM+Z, color( (-B_TERM)+Y, true ) )</var></code></p>
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<p>Isolate <code><var>X</var></code> by dividing both sides by its coefficient.</p>
<p><code><var>color( A_TERM-D_TERM, false )+X</var> = <var>plus( E_TERM+Z, (-B_TERM)+Y )</var></code></p>
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<p><code><var>X</var> = \dfrac{ <var>plus( E_TERM+Z, (-B_TERM)+Y )</var> }{ <var>color( A_TERM-D_TERM, false )</var> }</code></p>
<div data-if="GCD > 1 || (A_TERM-D_TERM) < 0" data-unwrap>
<p data-if="GCD > 1">All of these terms are divisible by <code><var>GCD</var></code>.</p>
<p data-if="GCD > 1 && (A_TERM-D_TERM) < 0">Divide by the common factor and swap signs so the denominator isn't negative.</p>
<p data-else-if="GCD > 1">Divide by the common factor.</p>
<p data-else>Swap signs so the denominator isn't negative.</p>
<p><code><var>X</var> = \dfrac{ <var>plus( color( round( E_TERM*DIVISOR ), true )+Z, color( round( -B_TERM*DIVISOR ), true )+Y )</var> }{ <var>color( round( (A_TERM-D_TERM)*DIVISOR ), true )</var> }</code></p>
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