// Numbas version: exam_results_page_options
{"name": "Simultaneous equations to solve forces in a system", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "rulesets": {"std": ["all"]}, "preamble": {"js": "", "css": ""}, "statement": "<p>The forces $F_1$ and $F_2$ in a system are described by the simultaneous equations given below.</p>\n<p>Solve for&nbsp;$F_1$ and $F_2$, giving your answers to $1$ decimal place where necessary.</p>\n<p><em>&nbsp;</em></p>", "question_groups": [{"name": "", "pickQuestions": 0, "questions": [], "pickingStrategy": "all-ordered"}], "tags": [], "extensions": [], "parts": [{"prompt": "<p>$\\simplify{{a}F_1 + {b}F_2 = {c}}$</p>\n<p>$\\simplify{{a1}F_1 + {b1}F_2 = {c1}}$</p>\n<p>$F_1$= &nbsp;[[0]]</p>\n<p>$F_2$= &nbsp;[[1]]</p>", "variableReplacements": [], "gaps": [{"variableReplacements": [], "scripts": {}, "showCorrectAnswer": true, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "precision": "1", "maxValue": "{ans1}+0.5", "allowFractions": false, "correctAnswerFraction": false, "showPrecisionHint": false, "type": "numberentry", "minValue": "{ans1}-0.5", "precisionPartialCredit": 0, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision."}, {"variableReplacements": [], "scripts": {}, "showCorrectAnswer": true, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "precision": "1", "maxValue": "{ans2} +0.5", "allowFractions": false, "correctAnswerFraction": false, "showPrecisionHint": false, "type": "numberentry", "minValue": "{ans2} -0.5", "precisionPartialCredit": 0, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision."}], "showCorrectAnswer": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "variables": {"ans1": {"description": "", "name": "ans1", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(1.1..2.5#0.1 except 2)"}, "num2": {"description": "", "name": "num2", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9 except num1)"}, "ans2": {"description": "", "name": "ans2", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(2.6..3.9#0.1 except 3)"}, "b": {"description": "", "name": "b", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(5..9 except [a])"}, "c": {"description": "", "name": "c", "group": "Ungrouped variables", "templateType": "anything", "definition": "(ans1*a)+(ans2*b)"}, "b1": {"description": "", "name": "b1", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(-8..9 except[a1,0])"}, "a1": {"description": "", "name": "a1", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)"}, "c1": {"description": "", "name": "c1", "group": "Ungrouped variables", "templateType": "anything", "definition": "(ans1*a1)+(ans2*b1)"}, "a": {"description": "", "name": "a", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(6..9)"}, "ans": {"description": "", "name": "ans", "group": "Ungrouped variables", "templateType": "anything", "definition": "precround((-num3 - (num1*num3))/((num1*num2)-num2),2)"}, "num1": {"description": "", "name": "num1", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)"}, "num3": {"description": "", "name": "num3", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9 except [num1,num2])"}}, "name": "Simultaneous equations to solve forces in a system", "ungrouped_variables": ["a", "b", "c", "a1", "b1", "c1", "ans1", "ans2", "num1", "num2", "num3", "ans"], "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "showQuestionGroupNames": false, "advice": "<p>Have a look at this youtube video to understand simultaneous equations better:&nbsp;<a href=\"https://www.youtube.com/watch?v=Lwto-lQzmec\">Simultaneous Equations</a></p>\n<p>Multiply the first equation by $\\var{b1}$ and the second equation by $\\var{b}$ so they both have the same $F_2$ coefficient:</p>\n<p>\\begin{align}<br/>\\simplify{{a*b1}F_1+{b*b1}F_2} &amp;= \\var{c*b1} \\\\<br/>\\simplify{{a1*b}F_1+{b1*b}F_2} &amp;= \\var{c1*b}<br/>\\end{align}</p>\n<p><i>(It is also fine to multiply the first equation by $~\\var{a1}$ and the second equation by $~\\var{a}$ in order to make the $F_1$ coefficients the same.)</i></p>\n<p>Then, subtract the second equation from the first to get</p>\n<p>\\[ \\simplify[std]{{a*b1-a1*b}F_1} = \\var{c*b1-c1*b} \\]</p>\n<p>So $F_1 = \\simplify[std]{{(c*b1-c1*b)/(a*b1-a1*b)}}$.</p>\n<p>Substitute this value of $F_1$ into the first (or second, but here we use the first) equation and rearrange to obtain $F_2$:</p>\n<p>\\begin{align}<br/>\\simplify[std]{{a}*{(c*b1-c1*b)/(a*b1-a1*b)} + {b}F_2} &amp;= \\var{c} \\\\<br/>\\simplify[std]{{b}F_2} &amp;= \\simplify[std]{{c}-{a*(c*b1-c1*b)/(a*b1-a1*b)}} \\\\<br/>F_2 &amp;= \\simplify[std]{{(c-a*(c*b1-c1*b)/(a*b1-a1*b))/b}}<br/>\\end{align}</p>", "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}]}]}], "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}]}