The student must solve a pair of simultaneous equations in $x$ and $y$.
\nThe variables are generated backwards: first $x$ and $y$ are picked, then values for the coefficients of the equations are chosen satisfying those values.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\\begin{align}
\\simplify{ {a}x + {b}y} &= \\var{c} \\\\
\\simplify{ {d}x + {e}y} &= \\var{f}
\\end{align}
Multiply the first equation by $\\var{e}$, and the second equation by $\\var{b}$, then subtract the second equation from the first:
\n\\begin{align}
\\simplify[basic, unitFactor]{ {e}*{a}x + {e}*{b}y } &= \\var{e} \\times \\var{c}\\\\
\\simplify[basic, unitFactor]{ {b}*{d}x + {b}*{e}y } &= \\var{b} \\times \\var{f} \\\\[1em]
\\simplify[basic, unitFactor]{ ({e}*{a} - {b}*{d})x + ({e}*{b} - {b}*{e})y } &= \\simplify[basic]{ {e*c} - {b*f}} \\\\[1em]
\\simplify{ {e*a-b*d}x } &= \\var{e*c-b*f} \\\\
x &= \\var{x}
\\end{align}
Then substitute this into the first equation and rearrange to find $y$:
\n\\begin{align}
\\simplify[basic, unitFactor]{ {a}*{x} + {b}y } &= \\var{c} \\\\
y &= \\var{y}
\\end{align}
The value of $x$ that satisfies the equations.
", "templateType": "randrange", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "random(-10 .. 10#1)", "description": "The value of $y$ that satisfies the equations.
", "templateType": "randrange", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "The coefficient of $x$ in the first equation.
", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "The coefficient of $y$ in the first equation.
", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "a*x + b*y", "description": "The right-hand side of the first equation.
", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "The coefficient of $x$ in the second equation.
", "templateType": "anything", "can_override": false}, "e": {"name": "e", "group": "Ungrouped variables", "definition": "random(-5..5 except [0, (d/a)*b])", "description": "The coefficient of $y$ in the second equation. There is one value which would make the second equation equivalent to the first: when $e = \\frac{d}{a} \\times b$, so that is excluded from the range of possible values.
", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "d*x + e*y", "description": "The right-hand side of the second equation.
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x", "y", "a", "b", "c", "d", "e", "f"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Solve the simultaneous equations.
\n$x = $ [[0]]
\n$y = $ [[1]]
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