// Numbas version: finer_feedback_settings {"name": "Gareth's copy of Denis's copy of Simultaneous equations (2)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"c1": {"templateType": "anything", "description": "", "definition": "random(1..5)", "group": "Ungrouped variables", "name": "c1"}, "c": {"templateType": "anything", "description": "", "definition": "random(1..9)", "group": "Ungrouped variables", "name": "c"}, "b": {"templateType": "anything", "description": "", "definition": "random(-9..9 except [0,a])", "group": "Ungrouped variables", "name": "b"}, "a1": {"templateType": "anything", "description": "", "definition": "random(1..9)", "group": "Ungrouped variables", "name": "a1"}, "b1": {"templateType": "anything", "description": "", "definition": "random(1..9 except round(a1*b/a))", "group": "Ungrouped variables", "name": "b1"}, "a": {"templateType": "anything", "description": "", "definition": "random(1..9)", "group": "Ungrouped variables", "name": "a"}}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "
Shows how to define variables to stop degenerate examples.
"}, "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}
Next, subtract the second equation from the first to get
\n\\[ \\simplify[std]{{a*b1-a1*b}x} = \\var{c*b1-c1*b} \\]
\nSo $x = \\simplify[std]{{(c*b1-c1*b)/(a*b1-a1*b)}}$.
\nSubstitute 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}
Solve the following pair of simultaneous equations:
\n\\[\\begin{eqnarray*} \\simplify{{a}x+{b}y}&=&\\var{c}\\\\\\\\\\simplify{{a1}x+{b1}y}&=&\\var{c1}\\end{eqnarray*}\\]
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\n$y=$ [[1]]
\ninput your answers as fractions in the form a/b and not as decimals.
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", "strings": ["."]}, "checkvariablenames": false, "answer": "{-c*a1+c1*a}/{a*b1-a1*b}", "answersimplification": "Std", "showCorrectAnswer": true, "vsetrangepoints": 5, "checkingaccuracy": 0.001, "checkingtype": "absdiff", "marks": 1, "type": "jme"}], "type": "gapfill", "marks": 0, "showCorrectAnswer": true}], "tags": [], "preamble": {"css": "", "js": ""}, "type": "question", "contributors": [{"name": "Gareth Woods", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/978/"}]}]}], "contributors": [{"name": "Gareth Woods", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/978/"}]}