// Numbas version: exam_results_page_options {"name": "Simultaneous equations by substitution 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "ans1", "ans2", "ans3", "yCoef", "n1", "n2", "n3", "n4"], "name": "Simultaneous equations by substitution 2", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "
We are going to solve for $x$ first. To do this, we need to eliminate $y$ from the equations.
\nWe start by rearranging equation (1) like so:
\n$\\simplify{y = 1/{n2}*({ans1}-{n1}*x)}$
\n$\\simplify{{n3}*x + {n4}/{n2}({ans1}-{n1}*x) = {ans2}}$
\n$\\simplify{({n3}-{n1}*{n4}/{n2})*x + {n4}*{ans1}/{n2} = {ans2}}$
\n$\\simplify{x=({n4}*{ans1}-{n2}*{ans2})/({n1}*{n4}-{n3}*{n2})}$
\nSubstitute the value of x back into equation (1) to find
\n$\\simplify{y=({ans1}*{n3}-{n1}*{ans2})/({n2}*{n3}-{n1}*{n4})}$
\n", "rulesets": {}, "parts": [{"prompt": "Solve the pair of equations
\n\\[\\begin{eqnarray} \\simplify{{n1}*x + {n2}*y} & = & \\var{ans1} &&&&&&&(1)\\\\ \\simplify{{n3}*x + {n4}*y} & = & \\var{ans2}&&&&&&&(2)\\end{eqnarray}\\]
\n\nWe are going to solve for $x$ first. To do this, we need to eliminate $y$ from the equations.
\nWe start by rearranging equation (1) like so:
\n$y = $[[0]], (3)
\nSubstitute equation (3) into equation (2) to give:
\n[[1]]$\\simplify{x + {n4}*{ans1}/{n2} = {ans2}}$
\nSolve this linear equation to give $x = $[[2]],
\nSubstitute the value of x back into equation (1) to find
\n$y = $[[3]]
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