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equation_of_a_line.html
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equation_of_a_line.html
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<!DOCTYPE html>
<html data-require="math math-format graphie">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Equation of a line</title>
<script src="../khan-exercise.js"></script>
<style>
.reading span {
width: 40px;
}
#answer_area .short input[type=text] {
width: 60px;
}
</style>
</head>
<body>
<div class="exercise">
<div class="vars" data-ensure="Math.pow(Y1 - Y2, 2) + Math.pow(X1 - X2, 2) < 36 && X1 < X2 && -10 <= B && B <= 10">
<var id="X1">randRange(-9, 9)</var>
<var id="Y1">randRange(-9, 9)</var>
<var id="X2">randRange(-9, 9)</var>
<var id="Y2">randRange(-9, 9)</var>
<var id="M">(Y1 - Y2) / (X1 - X2)</var>
<var id="B">Y1 - M * X1</var>
</div>
<div class="problems">
<div id="show-table">
<div class="vars">
<var id="COORDS">
(function() {
var coords = [];
coords.push( [ X1, [ Y1, 1 ] ] );
coords.push( [ X2, [ Y2, 1 ] ] );
var xs = randRangeUnique( -10, 10, 5 );
for ( var i = 0; i < 5; i++ ) {
var x = xs[ i ];
if( x !== X1 && x !== X2 ) {
var denom = X1 - X2,
num = x * ( Y1 - Y2 ) + B * denom,
negative = ( num * denom < 0 ? -1 : 1 );
num = round( abs( num ) * negative );
denom = round( abs( denom ) );
coords.push( [ x, [ num / getGCD( num, denom ), denom / getGCD( num, denom )] ] );
}
}
return coords.sort( function(a, b) { return a[ 0 ] - b[ 0 ]; });
})()
</var>
</div>
<p>A line goes through the following points, and the equation of that line is written in <code>y = mx + b</code> form.</p>
<p class="question">What is the equation of the line?</p>
<div class="fake_header reading">
<span>x</span><span>y</span>
</div>
<div class="fake_row reading" data-each="COORDS as i, coord">
<span><var>coord[ 0 ]</var></span><span><var>coord[ 1 ][ 1 ] === 1 ? coord[ 1 ][ 0 ] : coord[ 1 ].join( "/" )</var></span>
</div>
<div class="solution" data-type="multiple">
<p class="short"><code>y =</code> <span class="sol"><var>M</var></span><code>\enspace\cdot\enspace x + </code>
<span class="sol"><var>B</var></span></p>
</div>
<div class="hints">
<div>
<p>We can plot all the points and the line that connects them.</p>
<div class="graphie">
graphInit({
range: 10,
scale: 20,
tickStep: 1,
labelStep: 1,
unityLabels: false,
labelFormat: function( s ) { return "\\small{" + s + "}"; },
axisArrows: "<->"
});
style({ stroke: BLUE, fill: BLUE });
line( [X1 - 19, Y1 - 19 * M], [X2 + 19, Y2 + 19 * M], {
stroke: BLUE
} );
jQuery.each( COORDS, function( i, coord ) {
circle( [ coord[ 0 ], coord[ 1 ][ 0 ] / coord[ 1 ][ 1 ] ], 3 / 20 );
});
</div>
</div>
<p>We can choose any two points to determine the equation of the line.</p>
<p>Let's choose <code>(<var>X1</var>, <var>Y1</var>)</code> and <code>(<var>X2</var>, <var>Y2</var>)</code>.</p>
<p>The equation for the slope is <code>m = \dfrac{y_2 - y_1}{x_2 - x_1}</code>.</p>
<div>
<p>Substitute both points.</p>
<p><code>m = \dfrac{<var>Y2</var> - <var>negParens(Y1)</var>}{<var>X2</var> - <var>negParens(X1)</var>} = <var>fractionReduce( Y2 - Y1, X2 - X1 )</var></code></p>
</div>
<p>
Writing the equation of the line, we have <code>y = <var>( M === -1 ? "-" : ( M === 1 ? "" : fractionReduce( Y2 - Y1, X2 - X1 )))</var> x + b</code>
<span data-if="abs( M ) === 1"> (the value of <code>m</code> is equal to <code><var>M</var></code>)</span>.
</p>
<p>To find <code>b</code>, we can substitute in either of the two points into the above equation. Let's go through both cases:</p>
<div>
<p>Using the first point <code>(<var>X1</var>, <var>Y1</var>)</code>, substitute <code>y = <var>Y1</var></code> and <code>x = <var>X1</var></code>:</p>
<p><code><var>Y1</var> = (<var>fractionReduce( Y2 - Y1, X2 - X1 )</var>)(<var>X1</var>) + b</code></p>
<p><code>b = <var>Y1</var> - <var>fractionReduce( X1 * ( Y2 - Y1 ), X2 - X1 )</var> = <var>fractionReduce( Y1 * (X2 - X1) - X1 * ( Y2 - Y1 ), X2 - X1 )</var></code></p>
</div>
<div>
<p>Using the second point <code>(<var>X2</var>, <var>Y2</var>)</code>, substitute <code>y = <var>Y2</var></code> and <code>x = <var>X2</var></code>:</p>
<p><code><var>Y2</var> = (<var>fractionReduce( Y2 - Y1, X2 - X1 )</var>)(<var>X2</var>) + b</code></p>
<p><code>b = <var>Y2</var> - <var>fractionReduce( X2 * ( Y2 - Y1 ), X2 - X1 )</var> = <var>fractionReduce( Y2 * (X2 - X1) - X2 * ( Y2 - Y1 ), X2 - X1 )</var></code></p>
</div>
<p>In both cases, the equation of the line is <code>y = <var>( M === -1 ? "-" : ( M === 1 ? "" : fractionReduce( Y2 - Y1, X2 - X1 )))</var> x + <var>fractionReduce( Y1 * (X2 - X1) - X1 * ( Y2 - Y1 ), X2 - X1 )</var></code><span data-if="abs( M ) === 1"> (the value of <code>m</code> is equal to <code><var>M</var></code>)</span>.</p>
</div>
</div>
<div id="show-points-and-or-graph">
<div class="vars">
<var id="SHOW_GRAPH">randRange( 0, 1 )</var>
</div>
<div class="question">
<p>The equation of the line through the points <code>(<var>X1</var>, <var>Y1</var>)</code> and <code>(<var>X2</var>, <var>Y2</var>)</code> is written in the form <code>y = mx + b</code>.</p>
<p>What is the equation of the line?</p>
<div class="graphie" data-if="SHOW_GRAPH">
graphInit({
range: 10,
scale: 20,
tickStep: 1,
labelStep: 1,
unityLabels: false,
labelFormat: function( s ) { return "\\small{" + s + "}"; },
axisArrows: "<->"
});
style({ stroke: BLUE, fill: BLUE });
line( [X1 - 19, Y1 - 19 * M], [X2 + 19, Y2 + 19 * M] );
circle( [X1, Y1], 3/20 );
circle( [X2, Y2], 3/20 );
</div>
</div>
<div class="solution" data-type="multiple">
<p class="short"><code>y =</code> <span class="sol"><var>M</var></span><code>\enspace\cdot\enspace x + </code>
<span class="sol"><var>B</var></span></p>
</div>
<div class="hints">
<div class="graphie" data-if="!SHOW_GRAPH">
graphInit({
range: 10,
scale: 20,
tickStep: 1,
labelStep: 1,
unityLabels: false,
labelFormat: function( s ) { return "\\small{" + s + "}"; },
axisArrows: "<->"
});
style({ stroke: BLUE, fill: BLUE });
line( [X1 - 19, Y1 - 19 * M], [X2 + 19, Y2 + 19 * M] );
circle( [X1, Y1], 3/20 );
circle( [X2, Y2], 3/20 );
</div>
<p>The equation for the slope is <code>m = \dfrac{y_2 - y_1}{x_2 - x_1}</code>.</p>
<div>
<p>Substitute both points.</p>
<p><code>m = \dfrac{<var>Y2</var> - <var>negParens(Y1)</var>}{<var>X2</var> - <var>negParens(X1)</var>} = <var>fractionReduce( Y2 - Y1, X2 - X1 )</var></code></p>
</div>
<p>
Writing the equation of the line, we have <code>y = <var>( M === -1 ? "-" : ( M === 1 ? "" : fractionReduce( Y2 - Y1, X2 - X1 )))</var> x + b</code>
<span data-if="abs( M ) === 1"> (the value of <code>m</code> is equal to <code><var>M</var></code>)</span>.
</p>
<p>To find <code>b</code>, we can substitute in either of the two points into the above equation. Let's go through both cases:</p>
<div>
<p>Using the first point <code>(<var>X1</var>, <var>Y1</var>)</code>, substitute <code>y = <var>Y1</var></code> and <code>x = <var>X1</var></code>:</p>
<p><code><var>Y1</var> = (<var>fractionReduce( Y2 - Y1, X2 - X1 )</var>)(<var>X1</var>) + b</code></p>
<p><code>b = <var>Y1</var> - <var>fractionReduce( X1 * ( Y2 - Y1 ), X2 - X1 )</var> = <var>fractionReduce( Y1 * (X2 - X1) - X1 * ( Y2 - Y1 ), X2 - X1 )</var></code></p>
</div>
<div>
<p>Using the second point <code>(<var>X2</var>, <var>Y2</var>)</code>, substitute <code>y = <var>Y2</var></code> and <code>x = <var>X2</var></code>:</p>
<p><code><var>Y2</var> = (<var>fractionReduce( Y2 - Y1, X2 - X1 )</var>)(<var>X2</var>) + b</code></p>
<p><code>b = <var>Y2</var> - <var>fractionReduce( X2 * ( Y2 - Y1 ), X2 - X1 )</var> = <var>fractionReduce( Y2 * (X2 - X1) - X2 * ( Y2 - Y1 ), X2 - X1 )</var></code></p>
</div>
<p>In both cases, the equation of the line is <code>y = <var>( M === -1 ? "-" : ( M === 1 ? "" : fractionReduce( Y2 - Y1, X2 - X1 )))</var> x + <var>fractionReduce( Y1 * (X2 - X1) - X1 * ( Y2 - Y1 ), X2 - X1 )</var></code><span data-if="abs( M ) === 1"> (the value of <code>m</code> is equal to <code><var>M</var></code>)</span>.</p>
</div>
</div>
</div>
</div>
</body>
</html>