// Numbas version: exam_results_page_options {"name": "Gareth's copy of Gareth's copy of Gareth's copy of Denis's copy of Simultaneous equations (2)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["a", "c", "b", "a1", "b1", "c1"], "preamble": {"js": "", "css": ""}, "functions": {}, "parts": [{"gaps": [{"type": "jme", "checkvariablenames": false, "checkingtype": "absdiff", "vsetrangepoints": 5, "variableReplacements": [], "marks": 1, "variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "answer": "{b1*c-b*c1}/{a*b1-a1*b}", "notallowed": {"partialCredit": 0, "showStrings": false, "message": "

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$x=$ [[0]]

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$y=$ [[1]]

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

"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "

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

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

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Next, subtract the second equation from the first to get

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

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

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

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\\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}

", "rulesets": {"std": ["All", "fractionnumbers"]}, "statement": "

Solve the following pair of simultaneous equations:

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