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34ddff1 @elizabethslavitt New exercise: systems of equations word problems
elizabethslavitt authored
1 <!DOCTYPE html>
cde9e9f @jeresig Re-run the exercises through the updated cleaner.
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2 <html data-require="math math-format expressions word-problems spin">
3 <head>
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4 <meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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5 <title>Systems of equations word problems</title>
ac1415e @spicyj Use requirejs for module loading
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6 <script data-main="../local-only/main.js" src="../local-only/require.js"></script>
34ddff1 @elizabethslavitt New exercise: systems of equations word problems
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7 </head>
8 <body>
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9 <div class="exercise">
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10 <div class="problems">
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11 <div id="substitution-x">
12 <div class="vars">
13 <var id="INDEX">randRange( 0, 2 )</var>
14 <var id="X">[randRange( 2, 4 ), randRange( 2, 4 ), randRange( 40, 50 )][INDEX]</var>
15 <var id="Y">[randRange( X + 2, 10 ), randRange( X + 2, 10 ), randRange( 30, 50 )][INDEX]</var>
16 <var id="CENT_1">randRange( 0, 1 )</var>
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17 <var id="CENT_2" data-ensure="( CENT_1 + CENT_2 ) !== 0">randRange( 0, 1 )</var>
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18 <var id="A1">[roundTo( 8, randRange( 5, 8 ) + CENT_1 * 0.5 ), roundTo( 8, randRange( 5, 8 ) + CENT_1 * 0.5 ), 1][INDEX]</var>
19 <var id="B1">[roundTo( 8, randRange( 2, 4 ) + CENT_2 * 0.5 ), roundTo( 8, randRange( 2, 4 ) + CENT_2 * 0.5 ), 1][INDEX]</var>
20 <var id="C1">roundTo( 8, A1 * X + B1 * Y )</var>
21 <var id="A2">1</var>
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22 <var id="B2" data-ensure="( A1 * B2 ) !== ( A2 * B1 )">[-1, -1, randRange( -4, -2 )][INDEX]</var>
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23 <var id="C2">roundTo( 8, A2 * X + B2 * Y )</var>
24 <var id="SIGN_1">A1 * C2 &gt; 0 ? "-" : "+"</var>
25 <var id="SIGN_2">B1 * Y &gt; 0 ? "-" : "+"</var>
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26 <var id="PROBLEM_1">[
27 $._("A group of adults and kids went to see a movie"),
28 $._("A few families took a trip to an amusement park together"),
29 ""
30 ][INDEX]</var>
31 <var id="PROBLEM_2">[
32 $._("Find the number of adults and kids in the group"),
33 $._("Find the number of adults and kids on the trip"),
34 ""
35 ][INDEX]</var>
36 <var id="UNIT_1">[
37 $._("adults"),
38 $._("adults"),
39 $._("home team fans")
40 ][INDEX]</var>
41 <var id="UNIT_2">[
42 $._("kids"),
43 $._("kids"),
44 $._("away team fans")
45 ][INDEX]</var>
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46 </div>
47 <div class="question">
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48 <div data-if="INDEX === 0 || INDEX === 1">
49 <p><var>PROBLEM_1</var>. Tickets cost <code>$<var>localeToFixed(A1, 2)</var></code> each for adults and <code>$<var>localeToFixed(B1, 2)</var></code> each for kids, and the group paid <code>$<var>localeToFixed(C1, 2)</var></code> in total. There were <code><var>abs(C2)</var></code> fewer adults than kids in the group.</p>
50 <p><var>PROBLEM_2</var>.</p>
51 </div>
52 <div data-else="">
53 <p><code><var>C1</var></code> people attended a baseball game. Everyone there was a fan of either the home team or the away team.
54 The number of home team fans was <code><var>abs(C2)</var></code> less than <code><var>abs(B2)</var></code> times the number of away team fans.</p>
55 <p>How many home team and away team fans attended the game?</p>
56 </div>
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57 </div>
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58 <div class="solution" data-type="multiple">
59 <p># of <var>UNIT_1</var> = <span class="sol short50" data-forms="integer, decimal"><var>X</var></span></p>
60 <p># of <var>UNIT_2</var> = <span class="sol short50" data-forms="integer, decimal"><var>Y</var></span></p>
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61 </div>
62 <div class="hints">
63 <p>Let <code>x</code> equal the number of <var>UNIT_1</var> and <code>y</code> equal the number of <var>UNIT_2</var>.</p>
64 <div>
65 <p>The system of equations is then:</p>
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66 <p><code>\blue{<var>expr(["+", ["*", A1, "x"], ["*", B1, "y"]])</var> = <var>C1</var>}</code><br>
67 <code>\green{x = <var>expr(["+", ["*", -B2, "y"], C2])</var>}</code></p>
68 <p>Solve for <code>x</code> and <code>y</code> using substitution.</p>
69 </div>
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70 <div>
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71 <p>Since <code>x</code> has already been solved for, substitute <code>\green{<var>expr(["+", ["*", -B2, "y"], C2])</var>}</code> for <code>x</code> in the first equation.</p>
72 <p><code>\blue{<span data-if="abs(A1) !== 1"><var>A1</var></span><span data-if="A1 === -1">-</span>}\green{(<var>expr(["+", ["*", -B2, "y"], C2])</var>)}\blue{+ <var>expr(["*", B1, "y"])</var> = <var>C1</var>}</code></p>
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73 </div>
74 <div>
75 <p>Simplify and solve for <code>y</code>.</p>
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76 <p><code>
77 <var>expr(["+", ["*", roundTo(8, A1 * -B2), "y"], roundTo(8, A1 * C2)])</var> + <var>expr(["*", B1, "y"])</var> = <var>C1</var>
78 </code></p>
79 <p><code>
80 <var>expr(["+", ["*", roundTo(8, A1 * -B2 + B1), "y"], roundTo(8, A1 * C2)])</var> = <var>C1</var>
81 </code></p>
82 <p data-if="(A1 * -B2 + B1) !== 1"><code>
83 <var>expr(["*", roundTo(8, A1 * -B2 + B1), "y"])</var> = <var>roundTo(8, C1 - A1 * C2)</var>
84 </code></p>
85 <p data-if="(A1 * -B2 + B1) !== 1"><code>
86 y = \dfrac{<var>roundTo(8, C1 - A1 * C2)</var>}{<var>roundTo( 8, A1 * -B2 + B1 )</var>}
87 </code></p>
88 <p><code>\orange{y = <var>Y</var>}</code></p>
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89 </div>
90 <div>
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91 <p>Now that you know <code>\orange{y = <var>Y</var>}</code>, plug it back into <code>\green{x = <var>expr(["+", ["*", -B2, "y"], C2])</var>}</code> to find <code>x</code>.</p>
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92 </div>
93 <div>
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94 <p><code>\green{x = <span data-if="abs(B2) !== 1"><var>-B2</var></span><span data-if="B2 === 1">-</span>}\orange{(<var>Y</var>)}\green{ + <var>C2</var>}</code></p>
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95 <p data-if="B2 !== -1"><code>x = <var>-B2 * Y</var> + <var>C2</var></code></p>
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96 <p><code>\red{x = <var>X</var>}</code></p>
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97 </div>
98 <div>
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99 <p>You can also plug <code>\orange{y = <var>Y</var>}</code> into <code>\blue{<var>expr(["+", ["*", A1, "x"], ["*", B1, "y"]])</var> = <var>C1</var>}</code> and get the same answer for <code>x</code>:</p>
100 <p><code>\blue{<var>expr(["*", A1, "x"])</var> + <span data-if="abs(B1) !== 1"><var>B1</var></span><span data-if="B1 === -1">-</span>}\orange{(<var>Y</var>)}\blue{= <var>C1</var>}</code></p>
101 <p><code>\red{x = <var>X</var>}</code></p>
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102 </div>
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103 <p>There were <code><var>X</var></code> <var>UNIT_1</var> and <code><var>Y</var></code> <var>UNIT_2</var>.</p>
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104 </div>
105 </div>
106 <div id="substitution-y">
107 <div class="vars">
108 <var id="INDEX">randRange( 0, 2 )</var>
109 <var id="X">[randRange( 2, 4 ), randRange( 2, 4 ), randRange( 40, 50 )][INDEX]</var>
110 <var id="Y">[randRange( X + 2, 10 ), randRange( X + 2, 10 ), randRange( 30, 50 )][INDEX]</var>
111 <var id="CENT_1">randRange( 0, 1 )</var>
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112 <var id="CENT_2" data-ensure="( CENT_1 + CENT_2 ) !== 0">randRange( 0, 1 )</var>
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113 <var id="A1">[roundTo( 8, randRange( 5, 8 ) + CENT_1 * 0.5 ), roundTo( 8, randRange( 5, 8 ) + CENT_1 * 0.5 ), 1][INDEX]</var>
114 <var id="B1">[roundTo( 8, randRange( 2, 4 ) + CENT_2 * 0.5 ), roundTo( 8, randRange( 2, 4 ) + CENT_2 * 0.5 ), 1][INDEX]</var>
115 <var id="C1">roundTo( 8, A1 * X + B1 * Y )</var>
116 <var id="B2">1</var>
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117 <var id="A2" data-ensure="( A1 * B2 ) !== ( A2 * B1 )">[-1, -1, randRange( -4, -2 )][INDEX]</var>
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118 <var id="C2">roundTo( 8, A2 * X + B2 * Y )</var>
119 <var id="SIGN_1">B1 * C2 &gt; 0 ? "-" : "+"</var>
120 <var id="SIGN_2">A1 * X &gt; 0 ? "-" : "+"</var>
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121 <var id="PROBLEM_1">[
122 $._("The drama club sold bags of candy and cookies to raise money for the spring show"),
123 $._("A local gift shop sold bags of candy and cookies for Halloween"),
124 ""
125 ][INDEX]</var>
126 <var id="PROBLEM_2">[
127 $._("Find the number of bags of candy and cookies sold by the drama club"),
128 $._("Find the number of bags of candy and cookies sold by the gift shop"),
129 ""
130 ][INDEX]</var>
131 <var id="UNIT_1">[
132 $._("bags of candy"),
133 $._("bags of candy"),
134 $._("angle 1")
135 ][INDEX]</var>
136 <var id="UNIT_2">[
137 $._("bags of cookies"),
138 $._("bags of cookies"),
139 $._("angle 2")
140 ][INDEX]</var>
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141 </div>
142 <div class="question">
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143 <div data-if="INDEX === 0 || INDEX === 1">
144 <p><var>PROBLEM_1</var>. Bags of candy cost <code>$<var>localeToFixed(A1, 2)</var></code>, and bags of cookies cost <code>$<var>localeToFixed(B1, 2)</var></code>, and sales equaled <code>$<var>localeToFixed(C1, 2)</var></code> in total. There were <code><var>C2</var></code> more bags of cookies than candy sold.</p>
145 <p><var>PROBLEM_2</var>.</p>
146 </div>
147 <div data-else="">
148 <p>The sum of two angles is <code><var>C1</var>^\circ</code>. Angle 2 is <code><var>abs(C2)</var>^\circ</code> smaller than <code><var>abs(A2)</var></code> times angle 1.</p>
149 <p>What are the measures of the two angles in degrees?</p>
150 </div>
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151 </div>
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152 <div class="solution" data-type="multiple">
153 <p><span data-if="INDEX !== 2"># of </span><var>UNIT_1</var> = <span class="sol short50" data-forms="integer, decimal"><var>X</var></span></p>
154 <p><span data-if="INDEX !== 2"># of </span><var>UNIT_2</var> = <span class="sol short50" data-forms="integer, decimal"><var>Y</var></span></p>
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155 </div>
156 <div class="hints">
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157 <p data-if="INDEX !== 2">
158 Let <code>x</code> equal the number of <var>UNIT_1</var> and <code>y</code> equal the number of <var>UNIT_2</var>.
159 </p><p data-else>
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160 Let <code>x</code> equal the measure of <var>UNIT_1</var> and <code>y</code> equal the measure of <var>UNIT_2</var>.</p>
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161 <div>
162 <p>The system of equations is then:</p>
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163 <p><code>\blue{<var>expr(["+", ["*", A1, "x"], ["*", B1, "y"]])</var> = <var>C1</var>}</code><br>
164 <code>\green{y = <var>expr(["+", ["*", -A2, "x"], C2])</var>}</code></p>
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165 </div>
166 <div>
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167 <p>Since we already have solved for <code>y</code> in terms of <code>x</code>, we can use substitution to solve for <code>x</code> and <code>y</code>.</p>
168 </div>
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169 <div>
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170 <p>Substitute <code>\green{<var>expr(["+", ["*", -A2, "x"], C2])</var>}</code> for <code>y</code> in the first equation.</p>
171 <p><code>\blue{<var>expr(["*", A1, "x"])</var> + <span data-if="abs(B1) !== 1"><var>B1</var></span><span data-if="B1 === -1">-</span>}\green{(<var>expr(["+", ["*", -A2, "x"], C2])</var>)}\blue{= <var>C1</var>}</code></p>
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172 </div>
173 <div>
174 <p>Simplify and solve for <code>x</code>.</p>
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175 <p><code>
176 <var>expr(["+", ["*", A1, "x"], ["*", roundTo(8, B1 * -A2), "x"]])</var> + <var>roundTo(8, B1 * C2)</var> = <var>C1</var>
177 </code></p>
178 <p><code>
179 <var>expr(["+", ["*", roundTo(8, A1 + B1 * -A2), "x"], roundTo(8, B1 * C2)])</var> = <var>C1</var>
180 </code></p>
181 <p data-if="(A1 + B1 * -A2) !== 1"><code>
182 <var>expr(["*", roundTo(8, A1 + B1 * -A2), "x"])</var> = <var>roundTo(8, C1 - B1 * C2)</var>
183 </code></p>
184 <p data-if="(A1 + B1 * -A2) !== 1"><code>
185 x = \dfrac{<var>roundTo( 8, C1 - B1 * C2 )</var>}{<var>roundTo(8, A1 + B1 * -A2)</var>}
186 </code></p>
187 <p><code>\red{x = <var>X</var>}</code></p>
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188 </div>
189 <div>
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190 <p>Now that you know <code>\red{x = <var>X</var>}</code>, plug it back into <code> \green{y = <var>expr(["+", ["*", -A2, "x"], C2])</var>}</code> to find <code>y</code>.</p>
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191 </div>
192 <div>
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193 <p><code>\green{y = <span data-if="abs(A2) !== 1"><var>-A2</var></span><span data-if="A2 === 1">-</span>}\red{(<var>X</var>)}\green{ + <var>C2</var>}</code></p>
194 <p data-if="A2 !== -1"><code>y = <var>roundTo(8, -A2 * X)</var> + <var>C2</var></code></p>
195 <p><code>\orange{y = <var>Y</var>}</code></p>
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196 </div>
197 <div>
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198 <p>You can also plug <code>\red{x = <var>X</var>}</code> into <code> \blue{<var>expr(["+", ["*", A1, "x"], ["*", B1, "y"]])</var> = <var>C1</var>}</code> and get the same answer for <code>y</code>:</p>
199 <p><code>\blue{<span data-if="abs(A1) !== 1"><var>A1</var></span><span data-if="A1 === -1">-</span>}\red{(<var>X</var>)}\blue{ + <var>expr(["*", B1, "y"])</var> = <var>C1</var>}</code></p>
200 <p><code>\orange{y = <var>Y</var>}</code></p>
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201 </div>
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202 <p data-if="INDEX !== 2">
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203 <code><var>X</var></code> bags of candy and <code><var>Y</var></code> bags of cookies were sold.
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204 </p><p data-else>
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205 The measure of angle 1 is <code><var>X</var>^\circ</code> and the measure of angle 2 is <code><var>Y</var>^\circ</code>.
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206 </p>
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207 </div>
208 </div>
209 <div id="elimination-x">
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210 <div class="vars" data-ensure="abs( B1 * MULT1 + B2 * MULT2 ) &gt; 0.0001">
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211 <var id="X">randRange( 2, 4 )</var>
212 <var id="Y">randRange( X + 2, 10 )</var>
213 <var id="INDEX">1</var>
214 <var id="CENT_1">randRange( 0, 1 )</var>
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215 <var id="CENT_2" data-ensure="( CENT_1 + CENT_2 ) !== 0">randRange( 0, 1 )</var>
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216 <var id="Z1">roundTo( 8, randRange( 100, 300 ) / 100 )</var>
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217 <var id="Z2" data-ensure="abs( Z2 ) &lt;= 6 &amp;&amp; Z2 !== 0">INDEX === 0 ? -1 * Z1 : randRangeWeighted( -4, 4, -1, 0 ) * Z1</var>
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218 <var id="Z3">roundTo( 8, randRange( 200, 600 ) / 100 )</var>
219 <var id="Z4">roundTo( 8, randRange( 200, 600 ) / 100 )</var>
220 <var id="Z5">roundTo( 8, randRange( 200, 600 ) / 100 )</var>
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221 <var id="Z6" data-ensure="( Z3 * Z6 ) !== ( Z4 * Z5 )">roundTo( 8, randRange( 200, 600 ) / 100 )</var>
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222 <var id="A1">roundTo( 8, randRange( 5, 8 ) + CENT_1 * 0.5 )</var>
223 <var id="B1">roundTo( 8, randRange( 2, 4 ) + CENT_2 * 0.5 )</var>
224 <var id="C1">roundTo( 8, A1 * X + B1 * Y )</var>
225 <var id="A2">A1 * randRange( 2, 4 )</var>
226 <var id="B2">roundTo( 8, randRange( 7, 12 ) + ( CENT_1 + CENT_2 ) * 0.5 )</var>
227 <var id="C2">roundTo( 8, A2 * X + B2 * Y )</var>
228 <var id="MULT1">[ 1, roundTo( 8, -A2 / A1 ), 1, A2 &gt; 0 &amp;&amp; A1 &lt; 0 ? A2 : -A2 ][ INDEX ]</var>
229 <var id="MULT2">[ 1, 1, roundTo( 8, -A1 / A2 ), A2 &gt; 0 &amp;&amp; A1 &lt; 0 ? -A1 : A1 ][ INDEX ]</var>
230 <var id="SIGN_1">B1 * Y &gt; 0 ? "-" : "+"</var>
231 <var id="SIGN_2">B2 * Y &gt; 0 ? "-" : "+"</var>
232 </div>
233 <div class="question spin">
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234 <p>All of the {3rd|4th|5th} grade teachers and students from <var>school(1)</var> went on a field trip to an {art|archaeology} museum. Tickets were <code>$<var>localeToFixed(A1, 2)</var></code> each for teachers and <code>$<var>localeToFixed(B1, 2)</var></code> each for students, and the group paid <code>$<var>localeToFixed(C1, 2)</var></code> in total.</p>
235 <p>{A few weeks later|The next month}, the same group visited a {science|natural history} museum where the tickets cost <code>$<var>localeToFixed(A2, 2)</var></code> each for teachers and <code>$<var>localeToFixed(B2, 2)</var></code> each for students, and the group paid <code>$<var>localeToFixed(C2, 2)</var></code> in total.</p>
236 <p>Find the number of teachers and students on the field trips.</p>
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237 </div>
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238 <div class="solution" data-type="multiple">
239 <p># of teachers = <span class="sol short50" data-forms="integer, decimal"><var>X</var></span></p>
240 <p># of students = <span class="sol short50" data-forms="integer, decimal"><var>Y</var></span></p>
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241 </div>
242 <div class="hints">
243 <p>Let <code>x</code> equal the number of teachers and <code>y</code> equal the number of students.</p>
244 <div>
245 <p>The system of equations is:</p>
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246 <p><code>\blue{<var>expr(["+", ["*", A1, "x"], ["*", B1, "y"]])</var> = <var>C1</var>}</code><br>
247 <code>\green{<var>expr(["+", ["*", A2, "x"], ["*", B2, "y"]])</var> = <var>C2</var>}</code></p>
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248 <p>Solve for <code>x</code> and <code>y</code> using elimination.</p>
249 </div>
250 <div data-if="INDEX !== 0">
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251 <p data-if="INDEX === 2">
252 Multiply the bottom equation by <code><var>MULT2</var></code>.
253 </p><p data-else-if="INDEX === 3">
254 Multiply the top equation by <code><var>MULT1</var></code> and the bottom equation by <code><var>MULT2</var></code>.
255 </p><p data-else>
256 Multiply the top equation by <code><var>MULT1</var></code>.
257 </p>
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258 <p><code>\blue{<var>expr(["+", ["*", A1 * MULT1, "x"], ["*", B1 * MULT1, "y"]])</var> = <var>C1 * MULT1</var>}</code><br>
259 <code>\green{<var>expr(["+", ["*", A2 * MULT2, "x"], ["*", B2 * MULT2, "y"]])</var> = <var>C2 * MULT2</var>}</code></p>
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260 </div>
261 <div>
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262 <p>Add the top and bottom equations together.</p>
263 <p data-if="(B1 * MULT1 + B2 * MULT2) !== 1"><code>
264 <var>expr(["*", roundTo(8, B1 * MULT1 + B2 * MULT2), "y"])</var> = <var>roundTo(8, C1 * MULT1 + C2 * MULT2)</var>
265 </code></p>
266 <p data-if="(B1 * MULT1 + B2 * MULT2) !== 1"><code>
267 y = \dfrac{<var>roundTo(8, C1 * MULT1 + C2 * MULT2)</var>}{<var>roundTo(8, B1 * MULT1 + B2 * MULT2)</var>}</code></p>
268 <p><code>\orange{y = <var>Y</var>}</code></p>
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269 </div>
270 <div>
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271 <p>Now that you know <code>\orange{y = <var>Y</var>}</code>, plug it back into <code> \blue{<var>expr(["+", ["*", A1, "x"], ["*", B1, "y"]])</var> = <var>C1</var>}</code> to find <code>x</code>.</p>
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272 </div>
273 <div>
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274 <p><code>\blue{<var>expr(["*", A1, "x"])</var> + <span data-if="abs(B1) !== 1"><var>B1</var></span><span data-if="B1 === -1">-</span>}\orange{(<var>Y</var>)}\blue{= <var>C1</var>}</code></p>
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275 <p data-if="B1 !== 1"><code><var>expr(["+", ["*", A1, "x"], B1 * Y])</var> = <var>C1</var></code></p>
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276 <p data-if="A1 !== 1"><code><var>expr(["*", A1, "x"])</var> = <var>roundTo(8, C1 - B1 * Y)</var></code></p>
277 <p data-if="A1 !== 1"><code>x = \dfrac{<var>roundTo( 8, C1 - B1 * Y )</var>}{<var>A1</var>}</code></p>
278 <p><code>\red{x = <var>X</var>}</code></p>
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279 </div>
280 <div>
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281 <p>You can also plug <code>\orange{y = <var>Y</var>}</code> into <code> \green{<var>expr(["+", ["*", A2, "x"], ["*", B2, "y"]])</var> = <var>C2</var>}</code> and get the same answer for <code>x</code>:</p>
282 <p><code>\green{<var>expr(["*", A2, "x"])</var> + <span data-if="abs(B2) !== 1"><var>B2</var></span><span data-if="B2 === -1">-</span>}\orange{(<var>Y</var>)}\green{= <var>C2</var>}</code></p>
283 <p><code>\red{x = <var>X</var>}</code></p>
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284 </div>
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285 <p>There were <code><var>X</var></code> teachers and <code><var>Y</var></code> students on the field trips.</p>
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286 </div>
287 </div>
288 <div id="elimination-y">
289 <div class="vars">
290 <var id="X">randRange( 11, 99 )</var>
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291 <var id="Y" data-ensure="X !== Y">randRange( 10, X - 1 )</var>
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292 <var id="INDEX">0</var>
293 <var id="Z1">roundTo( 8, randRange( 100, 300 ) / 100 )</var>
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294 <var id="Z2" data-ensure="abs( Z2 ) &lt;= 6 &amp;&amp; Z2 !== 0">INDEX === 0 ? -1 * Z1 : randRangeWeighted( -4, 4, -1, 0 ) * Z1</var>
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295 <var id="Z3">roundTo( 8, randRange( 200, 600 ) / 100 )</var>
296 <var id="Z4">roundTo( 8, randRange( 200, 600 ) / 100 )</var>
297 <var id="Z5">roundTo( 8, randRange( 200, 600 ) / 100 )</var>
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298 <var id="Z6" data-ensure="( Z3 * Z6 ) !== ( Z4 * Z5 )">roundTo( 8, randRange( 200, 600 ) / 100 )</var>
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299 <var id="A1">1</var>
300 <var id="B1">[ 1, Z1, Z2, Z5 ][ INDEX ]</var>
301 <var id="C1">roundTo( 8, A1 * X + B1 * Y )</var>
302 <var id="A2">1</var>
303 <var id="B2">[ -1, Z2, Z1, Z6 ][ INDEX ]</var>
304 <var id="C2">roundTo( 8, A2 * X + B2 * Y )</var>
305 <var id="MULT1">[ 1, roundTo( 8, -B2 / B1 ), 1, B2 &gt; 0 &amp;&amp; B1 &lt; 0 ? B2 : -B2 ][ INDEX ]</var>
306 <var id="MULT2">[ 1, 1, roundTo( 8, -B1 / B2 ), B2 &gt; 0 &amp;&amp; B1 &lt; 0 ? -B1 : B1 ][ INDEX ]</var>
307 <var id="SIGN_1">A1 * X &gt; 0 ? "-" : "+"</var>
308 <var id="SIGN_2">A2 * X &gt; 0 ? "-" : "+"</var>
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309 <var id="LARGER">max(X, Y)</var>
310 <var id="SMALLER">min(X, Y)</var>
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311 </div>
312 <div class="question">
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313 <p>The sum of two numbers is <code><var>C1</var></code>, and their difference is <code><var>C2</var></code>. What are the two numbers?</p>
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314 </div>
7fab1d8 @spicyj Whitespace
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315
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316 <div class="solution" data-type="multiple">
317 <p>larger number = <span class="sol short50" data-forms="integer, decimal"><var>LARGER</var></span></p>
318 <p>smaller number = <span class="sol short50" data-forms="integer, decimal"><var>SMALLER</var></span></p>
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319 </div>
320 <div class="hints">
321 <div>
322 <p>Let <code>x</code> be the first number, and let <code>y</code> be the second number.</p>
323 </div>
324 <div>
325 <p>The system of equations is:</p>
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326 <p><code>\blue{<var>expr(["+", ["*", A1, "x"], ["*", B1, "y"]])</var> = <var>C1</var>}</code><br>
327 <code>\green{<var>expr(["+", ["*", A2, "x"], ["*", B2, "y"]])</var> = <var>C2</var>}</code></p>
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328 <p>Solve for <code>x</code> and <code>y</code> using elimination.
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329 </p></div>
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330 <div data-if="INDEX !== 0">
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331 <p data-if="INDEX === 2">
332 Multiply the bottom equation by <code><var>MULT2</var></code>.
333 </p><p data-else-if="INDEX === 3">
334 Multiply the top equation by <code><var>MULT1</var></code> and the bottom equation by <code><var>MULT2</var></code>.
335 </p><p data-else>
336 Multiply the top equation by <code><var>MULT1</var></code>.
337 </p>
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338 <p><code>\blue{<var>expr(["+", ["*", A1 * MULT1, "x"], ["*", B1 * MULT1, "y"]])</var> = <var>C1 * MULT1</var>}</code><br>
339 <code>\green{<var>expr(["+", ["*", A2 * MULT2, "x"], ["*", B2 * MULT2, "y"]])</var> = <var>C2 * MULT2</var>}</code></p>
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340 </div>
341 <div>
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342 <p>Add the top and bottom equations together.</p>
343 <p data-if="(A1 * MULT1 + A2 * MULT2) !== 1"><code>
344 <var>expr(["*", roundTo(8, A1 * MULT1 + A2 * MULT2), "x"])</var> = <var>roundTo(8, C1 * MULT1 + C2 * MULT2)</var>
345 </code></p>
346 <p data-if="(A1 * MULT1 + A2 * MULT2) !== 1"><code>
347 x = \dfrac{<var>roundTo(8, C1 * MULT1 + C2 * MULT2)</var>}{<var>roundTo( 8, A1 * MULT1 + A2 * MULT2 )</var>}
348 </code></p>
349 <p><code>\red{x = <var>X</var>}</code></p>
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350 </div>
351 <div>
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352 <p>Now that you know <code>\red{x = <var>X</var>}</code>, plug it back into <code> \blue{<var>expr(["+", ["*", A1, "x"], ["*", B1, "y"]])</var> = <var>C1</var>}</code> to find <code>y</code>.</p>
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353 </div>
354 <div>
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355 <p><code>\blue{<span data-if="abs(A1) !== 1"><var>A1</var></span><span data-if="A1 === -1">-</span>}\red{(<var>X</var>)}\blue{ + <var>expr(["*", B1, "y"])</var> = <var>C1</var>}</code></p>
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356 <p data-if="A1 !== 1"><code><var>expr(["+", A1 * X, ["*", B1, "y"]])</var> = <var>C1</var></code></p>
357 <p data-if="B1 !== 1"><code><var>expr(["*", B1, "y"])</var> = <var>roundTo( 8, C1 - A1 * X )</var></code></p>
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358 <p data-if="B1 !== 1"><code>\dfrac{<var>expr(["*", B1, "y"])</var>}{\blue{<var>B1</var>}} = \dfrac{<var>roundTo( 8, C1 - A1 * X )</var>}{\blue{<var>B1</var>}}</code></p>
359 <p><code>\orange{y = <var>Y</var>}</code></p>
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360 </div>
361 <div>
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362 <p>You can also plug <code>\red{x = <var>X</var>}</code> into <code> \green{<var>expr(["+", ["*", A2, "x"], ["*", B2, "y"]])</var> = <var>C2</var>}</code> and get the same answer for <code>y</code>:</p>
363 <p><code>\green{<span data-if="abs( A2 ) !== 1"><var>A2</var></span><span data-if="A2 === -1">-</span>}\red{(<var>X</var>)}\green{ + <var>expr(["*", B2, "y"])</var> = <var>C2</var>}</code></p>
364 <p><code>\orange{y = <var>Y</var>}</code></p>
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365 </div>
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366 <p>Therefore, the larger number is <code><var>LARGER</var></code>, and the smaller number is <code><var>SMALLER</var></code>.</p>
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367 </div>
368 </div>
369 </div>
370 </div>
cde9e9f @jeresig Re-run the exercises through the updated cleaner.
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371 </body>
372 </html>
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