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{"name": "Algebra VIII: solving simultaneous equations (rearrange & substitute)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "tags": [], "extensions": [], "functions": {}, "variables": {"ans2": {"definition": "{n3}*{a}+{n4}*{b}", "name": "ans2", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "n3": {"definition": "random(-10..10 except 0 except n1)", "name": "n3", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "n4": {"definition": "random(-10..10 except 0 except n2)", "name": "n4", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "n2": {"definition": "random(-10..10 except 0)", "name": "n2", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "yCoef": {"definition": "{n3}*{n2} - {n1}*{n4}", "name": "yCoef", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ans1": {"definition": "{n1}*{a} + {n2}*{b}", "name": "ans1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ans3": {"definition": "{ans1}*{n3} -  {ans2}*{n1}", "name": "ans3", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "n5": {"definition": "{ans2}- {n4}*{ans1}/{n2}", "name": "n5", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "n1": {"definition": "random(-10..10 except 0)", "name": "n1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b": {"definition": "random(-10..10 except 0 except a)", "name": "b", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a": {"definition": "random(-10..10 except 0)", "name": "a", "group": "Ungrouped variables", "templateType": "anything", "description": ""}}, "question_groups": [{"questions": [], "pickingStrategy": "all-ordered", "pickQuestions": 0, "name": ""}], "type": "question", "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "ans1", "ans2", "ans3", "yCoef", "n1", "n2", "n3", "n4", "n5"], "rulesets": {}, "parts": [{"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "prompt": "<p>We are going to solve for $x$ first. To do this, we need to eliminate $y$ from the equations.</p>\n<p>We start by rearranging equation (1)&nbsp;like so:</p>\n<p>$y =&nbsp;$[[0]]&nbsp; &nbsp; &nbsp; &nbsp;(3)</p>\n<p></p>\n<p></p>\n<p></p>", "gaps": [{"answersimplification": "all", "variableReplacements": [], "showpreview": true, "scripts": {}, "type": "jme", "expectedvariablenames": [], "answer": "1/{n2}*({ans1}-{n1}*x)", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "checkingtype": "absdiff", "checkvariablenames": false, "vsetrange": [0, 1], "marks": 1, "checkingaccuracy": 0.001, "vsetrangepoints": 5}], "marks": 0, "type": "gapfill"}, {"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "prompt": "<p>Substitute equation (3) into equation (2) to give:</p>\n<p>[[0]]$\\simplify[all,fractionnumbers]{x &nbsp;= {n5}}$</p>\n<p></p>", "gaps": [{"answersimplification": "all", "variableReplacements": [], "showpreview": true, "scripts": {}, "type": "jme", "expectedvariablenames": [], "answer": "{n3}-{n1}*{n4}/{n2}", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "checkingtype": "absdiff", "checkvariablenames": false, "vsetrange": [0, 1], "marks": 1, "checkingaccuracy": 0.001, "vsetrangepoints": 5}], "marks": 0, "type": "gapfill"}, {"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "prompt": "<p>Solve this linear equation to give &nbsp;</p>\n<p>$x = $[[0]]</p>\n<p></p>", "gaps": [{"answersimplification": "all", "variableReplacements": [], "showpreview": true, "scripts": {}, "type": "jme", "expectedvariablenames": [], "answer": "({n4}*{ans1}-{n2}*{ans2})/({n1}*{n4}-{n3}*{n2})", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "checkingtype": "absdiff", "checkvariablenames": false, "vsetrange": [0, 1], "marks": 1, "checkingaccuracy": 0.001, "vsetrangepoints": 5}], "marks": 0, "type": "gapfill"}, {"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "prompt": "<p>Substitute the value of x back into equation (1) to find</p>\n<p>$y = $[[0]]</p>", "gaps": [{"answersimplification": "all", "variableReplacements": [], "showpreview": true, "scripts": {}, "type": "jme", "expectedvariablenames": [], "answer": "({ans1}*{n3}-{n1}*{ans2})/({n2}*{n3}-{n1}*{n4})", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "checkingtype": "absdiff", "checkvariablenames": false, "vsetrange": [0, 1], "marks": 1, "checkingaccuracy": 0.001, "vsetrangepoints": 5}], "marks": 0, "type": "gapfill"}], "name": "Algebra VIII: solving simultaneous equations (rearrange & substitute)", "advice": "<p>We are going to solve for $x$ first. To do this, we need to eliminate $y$ from the equations.</p>\n<p>We start by rearranging equation (1)&nbsp;like so:</p>\n<p>$\\simplify{y = 1/{n2}*({ans1}-{n1}*x)}$</p>\n<p>$\\simplify{{n3}*x + {n4}/{n2}({ans1}-{n1}*x) = {ans2}}$</p>\n<p>$\\simplify{({n3}-{n1}*{n4}/{n2})*x + {n4}*{ans1}/{n2} = {ans2}}$</p>\n<p>$\\simplify{x=({n4}*{ans1}-{n2}*{ans2})/({n1}*{n4}-{n3}*{n2})}$</p>\n<p>Substitute the value of x back into equation (1) to find</p>\n<p>$\\simplify{y=({ans1}*{n3}-{n1}*{ans2})/({n2}*{n3}-{n1}*{n4})}$</p>\n<p></p>", "preamble": {"css": "", "js": ""}, "showQuestionGroupNames": false, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Straightforward solving linear equations question.</p>\n<p><em>Adapted from&nbsp;</em><strong>'Simultaneous equations by substitution 2 with parts'</strong><em> by&nbsp;</em><strong>Joshua Boddy</strong>.</p>"}, "statement": "<p>Solve the pair of simulatenous equations by working through parts a) to d)</p>\n<p>\\[\\begin{eqnarray} \\simplify{{n1}*x + {n2}*y} &amp; = &amp; \\var{ans1} &amp;&amp;&amp;&amp;&amp;&amp;&amp;(1)\\\\ \\simplify{{n3}*x + {n4}*y} &amp; = &amp; \\var{ans2}&amp;&amp;&amp;&amp;&amp;&amp;&amp;(2)\\end{eqnarray}\\]</p>", "contributors": [{"name": "Jean jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/353/"}]}]}], "contributors": [{"name": "Jean jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/353/"}]}