// Numbas version: exam_results_page_options {"name": "Viet's copy of Simultaneous equations by elimination 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["a", "b", "ans1", "ans2", "ans3", "yCoef", "n1", "n2", "n3", "n4"], "variables": {"b": {"templateType": "anything", "definition": "random(-10..10 except 0 except a)", "description": "", "name": "b", "group": "Ungrouped variables"}, "ans3": {"templateType": "anything", "definition": "{ans1}*{n3} - {ans2}*{n1}", "description": "", "name": "ans3", "group": "Ungrouped variables"}, "ans2": {"templateType": "anything", "definition": "{n3}*{a}+{n4}*{b}", "description": "", "name": "ans2", "group": "Ungrouped variables"}, "a": {"templateType": "anything", "definition": "random(-10..10 except 0)", "description": "", "name": "a", "group": "Ungrouped variables"}, "n1": {"templateType": "anything", "definition": "random(-10..10 except 0)", "description": "", "name": "n1", "group": "Ungrouped variables"}, "n3": {"templateType": "anything", "definition": "random(-10..10 except 0 except n1)", "description": "", "name": "n3", "group": "Ungrouped variables"}, "ans1": {"templateType": "anything", "definition": "{n1}*{a} + {n2}*{b}", "description": "", "name": "ans1", "group": "Ungrouped variables"}, "yCoef": {"templateType": "anything", "definition": "{n3}*{n2} - {n1}*{n4}", "description": "", "name": "yCoef", "group": "Ungrouped variables"}, "n4": {"templateType": "anything", "definition": "random(-10..10 except 0 except n2)", "description": "", "name": "n4", "group": "Ungrouped variables"}, "n2": {"templateType": "anything", "definition": "random(-10..10 except 0)", "description": "", "name": "n2", "group": "Ungrouped variables"}}, "statement": "", "extensions": [], "tags": [], "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

\n

We are going to solve for $y$ first. To do this, we need to eliminate $x$ from the equations. The quickest way to do this is to multiply the first equation by the co-efficient of $x$ in the second equation (here $\\var{n3}$), and multiply the second equation by the co-efficient of $x$ in the first equation (here $\\var{n1}$). We then get the equations:

\n

\n

[[0]]$x +$[[1]]$y = \\simplify{{ans1}*{n3}}$

\n

[[2]]$x +$[[3]]$y = \\simplify{{ans2}*{n1}}$

\n

\n

We then subtract one new equation from the other to get:

\n

[[4]]$\\simplify{y = {ans3}}$

\n

Now we can work out $y$

\n

$y =$[[5]]

\n

and substitute this value back in to any of the previous equations to get the value for $x$.

\n

$\\simplify{{n1}*x}$ + [[6]] = $\\var{ans1}$

\n

which then solves to give $x =$[[7]].

\n

\n

Straightforward solving linear equations question

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

\n

We are going to solve for $y$ first. To do this, we need to eliminate $x$ from the equations. The quickest way to do this is to multiply the first equation by the co-efficient of $x$ in the second equation (here $\\var{n3}$), and multiply the second equation by the co-efficient of $x$ in the first equation (here $\\var{n1}$). We then get the equations:

\n

\n

$\\simplify{{n3}*{n1}*x + {n3}*{n2}*y = {ans1}*{n3}}$

\n

$\\simplify{{n1}*{n3}*x + {n1}*{n4}*y = {ans2}*{n1}}$

\n

\n

We then subtract one new equation from the other to get:

\n

$\\simplify{{yCoef}y = {ans3}}$

\n

Now we can work out $y$

\n

$y = \\var{b}$

\n

and substitute this value back in to any of the previous equations to get the value for $x$.

\n

$\\simplify{{n1}*x + {n2}*{b} = {ans1}}$

\n

which then solves to give $x = \\var{a}$.

\n

\n

", "functions": {}, "preamble": {"js": "", "css": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": [], "name": "Viet's copy of Simultaneous equations by elimination 2", "rulesets": {}, "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}, {"name": "Viet Hoang", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2001/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}, {"name": "Viet Hoang", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2001/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}