// Numbas version: exam_results_page_options {"name": "Simultaneous equations (2)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"scripts": {}, "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "prompt": "

\\[\\begin{eqnarray*} \\simplify{{a}x+{b}y}&=&\\var{c}\\\\\\\\\\simplify{{a1}x+{b1}y}&=&\\var{c1}\\end{eqnarray*}\\]

\n

$x=$ [[0]]

\n

$y=$ [[1]]

\n

input your answers as fractions in the form a/b and not as decimals.

", "gaps": [{"variableReplacements": [], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "vsetrangepoints": 5, "notallowed": {"showStrings": false, "message": "

Input your answer as a fraction and not a decimal.

", "partialCredit": 0, "strings": ["."]}, "scripts": {}, "type": "jme", "checkvariablenames": false, "expectedvariablenames": [], "showFeedbackIcon": true, "answersimplification": "Std", "answer": "{b1*c-b*c1}/{a*b1-a1*b}", "showpreview": true, "checkingaccuracy": 0.001, "vsetrange": [0, 1], "checkingtype": "absdiff"}, {"variableReplacements": [], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "vsetrangepoints": 5, "notallowed": {"showStrings": false, "message": "

Input your answer as a fraction and not as a decimal.

", "partialCredit": 0, "strings": ["."]}, "scripts": {}, "type": "jme", "checkvariablenames": false, "expectedvariablenames": [], "showFeedbackIcon": true, "answersimplification": "Std", "answer": "{-c*a1+c1*a}/{a*b1-a1*b}", "showpreview": true, "checkingaccuracy": 0.001, "vsetrange": [0, 1], "checkingtype": "absdiff"}]}], "tags": [], "advice": "

Multiply the first equation by $\\var{b1}$ and the second equation by $\\var{b}$ so they both have the same $y$ coefficient:

\n

\\begin{align}
\\simplify{{a*b1}x+{b*b1}y} &= \\var{c*b1} \\\\
\\simplify{{a1*b}x+{b1*b}y} &= \\var{c1*b}
\\end{align}

\n

Next, subtract the second equation from the first to get

\n

\\[ \\simplify[std]{{a*b1-a1*b}x} = \\var{c*b1-c1*b} \\]

\n

So $x = \\simplify[std]{{(c*b1-c1*b)/(a*b1-a1*b)}}$.

\n

Substitute this value of $x$ into the first equation and rearrange to obtain $y$:

\n

\\begin{align}
\\simplify[std]{{a}*{(c*b1-c1*b)/(a*b1-a1*b)} + {b}y} &= \\var{c} \\\\
\\simplify[std]{{b}y} &= \\simplify[std]{{c}-{a*(c*b1-c1*b)/(a*b1-a1*b)}} \\\\
y &= \\simplify[std]{{(c-a*(c*b1-c1*b)/(a*b1-a1*b))/b}}
\\end{align}

", "name": "Simultaneous equations (2)", "preamble": {"js": "", "css": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Shows how to define variables to stop degenerate examples.

"}, "variable_groups": [], "functions": {}, "variables": {"b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9 except round(a1*b/a))", "description": "", "name": "b1"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "c"}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "a1"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "description": "", "name": "c1"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "a"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except [0,a])", "description": "", "name": "b"}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {"std": ["All", "fractionnumbers"]}, "extensions": [], "statement": "

Solve the following pair of simultaneous equations:

", "ungrouped_variables": ["a", "c", "b", "a1", "b1", "c1"], "type": "question", "contributors": [{"name": "Hannah Bartholomew", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/530/"}]}]}], "contributors": [{"name": "Hannah Bartholomew", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/530/"}]}