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{"name": "Ugur's copy of Simultaneous Equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Ugur's copy of Simultaneous Equations", "tags": [], "metadata": {"description": "<p>Exercises in solving simultaneous equations with 2 variables.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h2>Simultaneous Equations</h2>\n<p></p>\n<p>Solve the following simultaneous equations.</p>", "advice": "<p>To solve simultaneous equations, first&nbsp;rewrite the equations in matrix form of $Ab = C$, where the matrix $A$ will contain the coefficients of the equation, $b$ will contain the unknown values and $C$ will contain the constants.</p>\n<p>We rearrange&nbsp;$Ab = C$ to get $A^{-1}C = b$.</p>\n<p>So, we need to find the inverse of $A$ and multiple it with $C$. This will give us $b$, i.e. the values for $x$ and $y$.</p>\n<p></p>\n<p>For the first question:</p>\n<p>$A = \\simplify{{maA}}$&nbsp; $b = \\begin{pmatrix}x\\\\y\\end{pmatrix}$ $C = \\begin{pmatrix}\\var{a1}\\\\ \\var{a2}\\end{pmatrix}$</p>\n<p>To find the inverse of $A$, switch the elements on the leading diagonal, change the signs on the non-leading diagonal, and divide all elements by the determinant.</p>\n<p>The determinant of $A$ is $|A| = (\\var{maA[0][0]}\\times\\var{maA[1][1]}) -&nbsp;(\\var{maA[0][1]}\\times\\var{maA[1][0]}) = \\var{detA}$</p>\n<p>The inverse of $A$ is $A^{-1} = \\dfrac{1}{\\var{detA}}\\times\\begin{pmatrix}<br/>\\var{maAA[0][0]} &amp; \\var{maAA[0][1]} \\\\<br/>\\var{maAA[1][0]} &amp; \\var{maAA[1][1]}<br/>\\end{pmatrix} =<br/>\\begin{pmatrix}<br/>\\frac{\\var{maAA[0][0]}}{\\var{detA}} &amp; \\frac{\\var{maAA[0][1]}}{\\var{detA}} \\\\<br/>\\frac{\\var{maAA[1][0]}}{\\var{detA}} &amp;&nbsp;\\frac{\\var{maAA[1][1]}}{\\var{detA}}<br/>\\end{pmatrix}$</p>\n<p></p>\n<p>Rearranging for&nbsp;$A^{-1}C = b$:</p>\n<p>$A^{-1}\\times C =&nbsp;<br/>\\begin{pmatrix}<br/>\\frac{\\var{maAA[0][0]}}{\\var{detA}} &amp; \\frac{\\var{maAA[0][1]}}{\\var{detA}} \\\\<br/>\\frac{\\var{maAA[1][0]}}{\\var{detA}} &amp;&nbsp;\\frac{\\var{maAA[1][1]}}{\\var{detA}}<br/>\\end{pmatrix}<br/>\\times\\begin{pmatrix}\\var{a1}\\\\ \\var{a2}\\end{pmatrix} =<br/>\\begin{pmatrix}<br/>\\frac{\\var{maAA[0][0]}}{\\var{detA}}\\times\\var{a1} +&nbsp;\\frac{\\var{maAA[0][1]}}{\\var{detA}}\\times\\var{a2} \\\\<br/>\\frac{\\var{maAA[1][0]}}{\\var{detA}}\\times\\var{a1} +&nbsp;\\frac{\\var{maAA[1][1]}}{\\var{detA}}\\times\\var{a2}<br/>\\end{pmatrix} =<br/> \\begin{pmatrix}\\var{ax}\\\\ \\var{ay}\\end{pmatrix}$</p>\n<p>Therefore $x = \\var{ax}$ and&nbsp;$y = \\var{ay}$.</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": 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"description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["ax", "ay", "maA", "a1", "a2", "bx", "by", "maB", "b1", "b2", "maAA", "detA"], "variable_groups": [], "functions": {"matrixrandomN": {"parameters": [["rows", "integer"], ["columns", "integer"]], "type": "matrix", "language": "jme", "definition": "matrix(repeat(repeat(random(1..9)*(random(1,1,1,-1)),columns),rows))"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\simplify{{maA[0][0]}}x + \\simplify{{maA[0][1]}}y = 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