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{"name": "Algebra VIII: solving simultaneous equations (by elimination)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"css": "", "js": ""}, "variables": {"yCoef": {"templateType": "anything", "definition": "{n3}*{n2} - {n1}*{n4}", "name": "yCoef", "group": "Ungrouped variables", "description": ""}, "ans1": {"templateType": "anything", "definition": "{n1}*{a} + {n2}*{b}", "name": "ans1", "group": "Ungrouped variables", "description": ""}, "ans2": {"templateType": "anything", "definition": "{n3}*{a}+{n4}*{b}", "name": "ans2", "group": "Ungrouped variables", "description": ""}, "n1": {"templateType": "anything", "definition": "random(-10..10 except 0)", "name": "n1", "group": "Ungrouped variables", "description": ""}, "n3": {"templateType": "anything", "definition": "random(-10..10 except 0 except n1)", "name": "n3", "group": "Ungrouped variables", "description": ""}, "n4": {"templateType": "anything", "definition": "random(-10..10 except 0 except n2)", "name": "n4", "group": "Ungrouped variables", "description": ""}, "a": {"templateType": "anything", "definition": "random(-10..10 except 0)", "name": "a", "group": "Ungrouped variables", "description": ""}, "ans3": {"templateType": "anything", "definition": "{ans1}*{n3} -  {ans2}*{n1}", "name": "ans3", "group": "Ungrouped variables", "description": ""}, "b": {"templateType": "anything", "definition": "random(-10..10 except 0 except a)", "name": "b", "group": "Ungrouped variables", "description": ""}, "n2": {"templateType": "anything", "definition": "random(-10..10 except 0)", "name": "n2", "group": "Ungrouped variables", "description": ""}}, "advice": "<p>Solve the pair of equations</p>\n<p>\\[\\begin{eqnarray} \\simplify{{n1}*x + {n2}*y -{ans1}} = 0 &amp;&amp;&amp;&amp;&amp;&amp;&amp;(1)\\\\ \\simplify{{n3}*x + {n4}*y -{ans2}} = 0 &amp;&amp;&amp;&amp;&amp;&amp;&amp;(2)\\end{eqnarray}\\]</p>\n<p></p>\n<p>We are going to solve for $y$ first. To do this, we need to eliminate $x$ from the equations. The quickest way to do this is to multiply the first equation by the co-efficient of $x$ in the second equation (here $\\var{n3}$), and multiply the second equation by the co-efficient of $x$ in the first equation (here $\\var{n1}$). We then get the equations:</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: center;\">$\\simplify{{n3}*{n1}*x +&nbsp;{n3}*{n2}*y +{ans1}*(-1)*{n3}} = 0$</p>\n<p style=\"text-align: center;\">&nbsp;$\\simplify{{n1}*{n3}*x +&nbsp;{n1}*{n4}*y +{ans2}*(-1)*{n1}} = 0$</p>\n<p></p>\n<p>We then subtract one new equation from the other to get:</p>\n<p style=\"text-align: center;\">$\\simplify{{yCoef}y - {ans3} = 0}$</p>\n<p>Now we can work out $y$</p>\n<p style=\"text-align: center;\">$y = \\var{b}$&nbsp;</p>\n<p>and substitute this value back in to any of the previous equations to get the value for $x$.&nbsp;</p>\n<p style=\"text-align: center;\">$\\simplify{{n1}*x +&nbsp;{n2}*{b} + (-1)*{ans1}}$ = 0</p>\n<p style=\"text-align: center;\">which then solves to give $x = \\var{a}$.</p>\n<p></p>", "variable_groups": [], "rulesets": {}, "parts": [{"prompt": "<p>We are going to solve for $y$ first. To do this, we need to eliminate $x$ from the equations. The quickest way to do this is to multiply the first equation by the co-efficient of $x$ in the second equation (here $\\var{n3}$), and multiply the second equation by the co-efficient of $x$ in the first equation (here $\\var{n1}$). We then get the equations:</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: center;\">[[0]]$x$ + [[1]]$\\simplify{y +{ans1}*(-1)*{n3}} = 0$</p>\n<p style=\"text-align: center;\">&nbsp;[[2]]$x$ + [[3]]$\\simplify{y +{ans2}*(-1)*{n1}} = 0$</p>\n<p></p>", "marks": 0, "scripts": {}, "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "gaps": [{"expectedvariablenames": [], "checkingaccuracy": 0.001, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "answer": "{n3}*{n1}", "type": "jme", "vsetrangepoints": 5, "showCorrectAnswer": true, "showpreview": true, "checkingtype": "absdiff"}, {"expectedvariablenames": [], "checkingaccuracy": 0.001, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "answer": "{n3}*{n2}", "type": "jme", "vsetrangepoints": 5, "showCorrectAnswer": true, "showpreview": true, "checkingtype": "absdiff"}, {"expectedvariablenames": [], "checkingaccuracy": 0.001, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "answer": "{n1}*{n3}", "type": "jme", "vsetrangepoints": 5, "showCorrectAnswer": false, "showpreview": true, "checkingtype": "absdiff"}, {"expectedvariablenames": [], "checkingaccuracy": 0.001, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "answer": "{n1}*{n4}", "type": "jme", "vsetrangepoints": 5, "showCorrectAnswer": true, "showpreview": true, "checkingtype": "absdiff"}], "variableReplacementStrategy": "originalfirst"}, {"prompt": "<p style=\"text-align: left;\">We then subtract one new equation from the other to get:</p>\n<p style=\"text-align: center;\">[[0]]$\\simplify{y - {ans3} = 0}$</p>\n<p></p>", "marks": 0, "scripts": {}, "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "gaps": [{"expectedvariablenames": [], "checkingaccuracy": 0.001, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "answer": "{yCoef}", "type": "jme", "vsetrangepoints": 5, "showCorrectAnswer": true, "showpreview": true, "checkingtype": "absdiff"}], "variableReplacementStrategy": "originalfirst"}, {"prompt": "<p>Now we can work out $y$</p>\n<p style=\"text-align: center;\">$y =$[[0]]</p>\n<p></p>", "marks": 0, "scripts": {}, "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "gaps": [{"expectedvariablenames": [], "checkingaccuracy": 0.001, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "answer": "{b}", "type": "jme", "vsetrangepoints": 5, "showCorrectAnswer": true, "showpreview": true, "checkingtype": "absdiff"}], "variableReplacementStrategy": "originalfirst"}, {"prompt": "<p>and substitute this value back in to any of the previous equations to get the value for $x$.&nbsp;</p>\n<p style=\"text-align: center;\">$\\simplify{{n1}*x}$ +&nbsp;[[0]]&nbsp;$+\\simplify{(-1)*{ans1}}$ = 0</p>\n<p style=\"text-align: center;\"></p>", "marks": 0, "scripts": {}, "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "gaps": [{"expectedvariablenames": [], "checkingaccuracy": 0.001, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "answer": "{n2}*{b}", "type": "jme", "vsetrangepoints": 5, "showCorrectAnswer": true, "showpreview": true, "checkingtype": "absdiff"}], "variableReplacementStrategy": "originalfirst"}, {"prompt": "<p>which then solves to give $x = $[[0]].</p>", "marks": 0, "scripts": {}, "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "gaps": [{"expectedvariablenames": [], "checkingaccuracy": 0.001, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "answer": "{a}", "type": "jme", "vsetrangepoints": 5, "showCorrectAnswer": true, "showpreview": true, "checkingtype": "absdiff"}], "variableReplacementStrategy": "originalfirst"}], "tags": [], "statement": "<p>Solve the pair of simultaneous equations by working through parts a) to e)</p>\n<p>\\[\\begin{eqnarray} \\simplify{{n1}*x + {n2}*y -{ans1}} = 0 &amp;&amp;&amp;&amp;&amp;&amp;&amp;(1)\\\\ \\simplify{{n3}*x + {n4}*y -{ans2}} = 0 &amp;&amp;&amp;&amp;&amp;&amp;&amp;(2)\\end{eqnarray}\\]</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "question_groups": [{"pickQuestions": 0, "name": "", "questions": [], "pickingStrategy": "all-ordered"}], "showQuestionGroupNames": false, "type": "question", "extensions": [], "ungrouped_variables": ["a", "b", "ans1", "ans2", "ans3", "yCoef", "n1", "n2", "n3", "n4"], "name": "Algebra VIII: solving simultaneous equations (by elimination)", "functions": {}, "metadata": {"description": "<p>Straightforward solving linear equations question.</p>\n<p><em>Adapted from&nbsp;</em><strong>'Simultaneous equations by elimination 1&nbsp;with parts'</strong><em> by&nbsp;</em><strong>Joshua Boddy</strong>.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "contributors": [{"name": "Luke Park", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/826/"}, {"name": "heike hoffmann", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2960/"}]}]}], "contributors": [{"name": "Luke Park", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/826/"}, {"name": "heike hoffmann", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2960/"}]}