// Numbas version: exam_results_page_options
{"name": "Systems of linear equations in 2 variables", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "<p>Consider the following system of equations:</p>\n<p>\\begin{align}&nbsp;<br/>\\simplify{{a}x+{b}y}&amp;=\\var{c}\\text{,}\\\\<br/>\\simplify{{a1}x+{b1}y}&amp;=\\var{c1}\\text{.}<br/>\\end{align}</p>", "preamble": {"js": "", "css": ""}, "extensions": [], "ungrouped_variables": ["a", "c", "b", "a1", "b1", "c1"], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": ""}, "parts": [{"marks": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>Solve to find the values of $x$ and $y$.&nbsp;</p>\n<p>$x=$&nbsp;[[0]]</p>\n<p>$y=$&nbsp;[[1]]</p>\n<p></p>", "gaps": [{"correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacements": [], "allowFractions": true, "maxValue": "{b1*c-b*c1}/{a*b1-a1*b}+0.01", "notationStyles": ["plain", "en", "si-en"], "minValue": "{b1*c-b*c1}/{a*b1-a1*b}-0.01", "marks": 1, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "type": "numberentry", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerFraction": true}, {"correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacements": [], "allowFractions": true, "maxValue": "{-c*a1+c1*a}/{a*b1-a1*b}+0.01", "notationStyles": ["plain", "en", "si-en"], "minValue": "{-c*a1+c1*a}/{a*b1-a1*b}-0.01", "marks": 1, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "type": "numberentry", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerFraction": true}], "type": "gapfill", "showFeedbackIcon": true}], "name": "Systems of linear equations in 2 variables", "variables": {"a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,a])", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "c"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9 except round(a1*b/a))", "description": "", "name": "b1"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a1"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "c1"}}, "functions": {}, "tags": [], "variable_groups": [], "rulesets": {"std": ["All", "fractionnumbers"]}, "advice": "<p>Multiply the first equation by $\\var{b1}$ and the second equation by $\\var{b}$ so they both have the same $y$ coefficient:</p>\n<p>\\begin{align}<br/>\\simplify{{a*b1}x+{b*b1}y} &amp;= \\var{c*b1} \\\\<br/>\\simplify{{a1*b}x+{b1*b}y} &amp;= \\var{c1*b}<br/>\\end{align}</p>\n<p>Next, subtract the second equation from the first to get</p>\n<p>\\[ \\simplify[std]{{a*b1-a1*b}x} = \\var{c*b1-c1*b} \\]</p>\n<p>So $x = \\displaystyle \\simplify[std]{{(c*b1-c1*b)/(a*b1-a1*b)}}$.</p>\n<p>Substitute this value of $x$ into the first equation and rearrange to obtain $y$:</p>\n<p>\\begin{align}<br/>\\simplify[std]{{a}*{(c*b1-c1*b)/(a*b1-a1*b)} + {b}y} &amp;= \\var{c} \\\\<br/>\\simplify[std]{{b}y} &amp;= \\simplify[std]{{c}-{a*(c*b1-c1*b)/(a*b1-a1*b)}} \\\\<br/>y &amp;= \\simplify[std]{{(c-a*(c*b1-c1*b)/(a*b1-a1*b))/b}}<br/>\\end{align}</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question", "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}]}]}], "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}]}