// Numbas version: exam_results_page_options
{"name": "Maths Support: Solving linear equations", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "duration": 0.0, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "shufflequestions": false, "questions": [], "percentpass": 0.0, "allQuestions": true, "pickQuestions": 0, "type": "exam", "feedback": {"showtotalmark": true, "advicethreshold": 0.0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Solving equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["equations", "linear equation", "solving equations", "solving linear equations"], "advice": "\n    <p>Given the equation \\[\\simplify[std]{{a}x+{b}={c}x+{d}}\\] we first collect together all the constant terms, and collect together all the terms in $x$.</p>\n  <p>The equation can then be written as:<br/>\\[\\simplify[std]{({a}-{c})x=({d}+{-b})}\\] i.e.<br/>\\[\\simplify{{a-c}x={d-b}}\\]<br/>which gives \\[x =\\simplify[std]{{(d-b)}/{(a-c)}}\\] as the solution.</p>\n    ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "parts": [{"prompt": "\n        <p>\\[\\simplify[std]{{a} * x + {b} = {c} * x + {d}}\\]<br/>Input your answer as a fraction or an integer. Do <span class=\"caps\">NOT</span> input the answer as a decimal.</p>\n      <p>$x\\;=$[[0]]</p>\n        ", "gaps": [{"notallowed": {"message": "<p>Input your answer as a fraction or an integer. Do&nbsp;<span class=\"caps\">not</span>&nbsp;input the answer as a decimal.&nbsp;</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "all", "marks": 1.0, "answer": "{d-b}/{a-c}", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "statement": "<p>Solve the following linear equation for $x$.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "s1*random(2..12)", "name": "a"}, "c": {"definition": "if(a=tc,tc+1,tc)", "name": "c"}, "b": {"definition": "s2*random(1..12)", "name": "b"}, "d": {"definition": "if(b=td,td+1,td)", "name": "d"}, "s3": {"definition": "random(1,-1)", "name": "s3"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "td": {"definition": "random(1..20)", "name": "td"}, "tc": {"definition": "s3*random(2..20)", "name": "tc"}}, "metadata": {"notes": "\n        \t\t<p><strong>5/08/2012:</strong></p>\n        \t\t    <p>Added more tags.</p>\n        \t\t    <p>Added description.</p>\n        \t\t    <p>Checked calculation. OK.</p>\n        \t\t", "description": "<p>Solve $\\displaystyle ax + b = cx + d$ for $x$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Solving equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "changing the subject of an equation", "rearranging equations", "solving", "solving equations", "subject of an equation"], "advice": "<p>Rearrange the equation by adding {-c} to both sides to get:<br/>\\[\\simplify{{t} / ({a} * x + {b}) = {d} + { -c} = {d -c}}\\]<br/>This gives \\[\\simplify{({a} * x + {b}) / {t} = 1 / {d -c}}\\] (this is because if $\\displaystyle \\frac{a}{b}=c$ then $\\displaystyle \\frac{b}{a}=\\frac{1}{c}$ on turning the fraction round the other way)<br/>and so \\[\\simplify{({a} * x + {b}) = {t} / {d -c}}\\] on multiplying both sides by {t}.<br/>Hence \\[\\simplify{{a} * x = {t} / {d -c} -{b} = ({a * an1} / {an2})}\\]<br/>and so \\[\\simplify{x={an1}/{an2}}\\] is the solution on dividing both sides by {a}.</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n        <p>\\[\\simplify{{t} / ({a} * x + {b}) + {c} = {d}}\\]</p>\n      <p>$x=\\;$ [[0]]</p>\n      <p>If you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.</p>\n      <p>Input all numbers as fractions or integers and not as decimals.</p>\n        ", "gaps": [{"notallowed": {"message": "<p>Input as a fraction or an integer, not as a decimal.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.0001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 2.0, "answer": "{an1}/{an2}", "type": "jme"}], "steps": [{"prompt": "\n            <p>Rearrange the equation by adding {-c} to both sides to get:<br/>\\[\\simplify[std]{{t} / ({a} * x + {b}) = {d} + { -c} = {d -c}}\\]</p>\n          <p>This gives \\[\\simplify{({a} * x + {b}) / {t} = 1 / {d -c}}\\] (this is because if $\\displaystyle \\frac{a}{b}=c$ then $\\displaystyle \\frac{b}{a}=\\frac{1}{c}$ on turning the fraction round the other way)</p>\n          <p>and so \\[\\simplify{({a} * x + {b}) = {t} / {d -c}}\\] on multiplying both sides by {t}.</p>\n          <p>Solve this equation for $x$.</p>\n            ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n    <p>Solve the following equation for $x$.</p>\n  <p>Input your answer as a fraction or an integer as appropriate and not&nbsp;as a decimal.</p>\n    ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..9)", "name": "a"}, "c": {"definition": "s2*random(1..9)", "name": "c"}, "b": {"definition": "if(a=abs(b1),abs(b1)+2,b1)", "name": "b"}, "d": {"definition": "abs(c)+random(2..9)", "name": "d"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(-1,1)", "name": "s1"}, "t": {"definition": "random(2..8)", "name": "t"}, "an2": {"definition": "a*(d-c)", "name": "an2"}, "an1": {"definition": "t-b*d+b*c", "name": "an1"}, "b1": {"definition": "s1*random(1..10)", "name": "b1"}}, "metadata": {"notes": "\n        \t\t<p><strong>5/08/2012:</strong></p>\n        \t\t    <p>Added tags.</p>\n        \t\t    <p>Added description.</p>\n        \t\t    <p>Checked calculation.OK.</p>\n        \t\t    <p>Improved display in content areas.</p>\n        \t\t", "description": "<p>Solve for $x$: $\\displaystyle \\frac{a} {bx+c} + d= s$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Solving equations. (Video)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Solving equations", "equations", "linear equation", "solving a linear equation in one variable", "solving equations", "solving linear equations", "video"], "advice": "<p>Given the equation \\[\\simplify[std]{{a} * x + {b} = {f}/{g}({c} * x + {d})}\\] we first multiply both sides by $\\var{g}$ to get</p>\n  <p>\\[\\simplify[std]{{g}*({a} * x + {b} )= {f}*({c} * x + {d})}.\\]</p>\n  <p>Then expand both sides of this equation to get:</p>\n  <p>\\[\\simplify[std]{{g*a} x + {g*b} = {f*c}x + {f*d}}.\\]</p>\n  <p>and then&nbsp;collect together all the constant terms on the right hand-side, and collect together all the terms in $x$ on the left-hand side of the equation.</p>\n  <p>The equation can then be written as:<br/>\\[\\simplify[std]{({g*a}-{f*c})x=({f*d}+{-g*b})}\\] i.e.<br/>\\[\\simplify{{g*a-f*c}x={f*d-b*g}}\\]<br/>which gives \\[x =\\simplify[std]{{(f*d-b*g)}/{(g*a-f*c)}}\\] as the solution.</p>\n  <p><strong>Check the answer</strong></p>\n  <p>You can check that this is the correct solution by inputting this solution back into the equation to see if it satisfies the equation.&nbsp;</p>\n  <p>&nbsp;</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "parts": [{"stepspenalty": 0.0, "prompt": "<p>\\[\\simplify[std]{{a} * x + {b} = {f}/{g}({c} * x + {d})}\\]<br/>Input your answer as a fraction or an integer. Do <span class=\"caps\">NOT</span> input the answer as a decimal.</p>\n        <p>$x =$ [[0]]</p>\n        <p>Click on Show steps to see a video of a solution of a similar problem.</p>", "gaps": [{"notallowed": {"message": "<p>Input your answer as a fraction or an integer. Do&nbsp;<span class=\"caps\">not</span>&nbsp;input the answer as a decimal.&nbsp;</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "all", "marks": 1.0, "answer": "{f*d-b*g}/{g*a-f*c}", "checkvariablenames": false, "type": "jme"}], "steps": [{"prompt": "<p>A video example worked through. The method in the video is slightly different from the method in the solution.</p>\n          <p><iframe width=\"640\" height=\"480\" src=\"https://player.vimeo.com/video/8173488\" frameborder=\"0\"></iframe></p>", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "<p>Solve the following linear equation for $x$.</p>\n    <p>&nbsp;</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(-12..12)", "name": "a"}, "c": {"definition": "if(a*g=tc*f,tc+1,tc)", "name": "c"}, "b": {"definition": "s2*random(1..12)", "name": "b"}, "d": {"definition": "if(b=td,td+1,td)", "name": "d"}, "g": {"definition": "switch(f=1,random(2..5),f=2, random(3..5),f=3, random(2,4,5),f=4,random(3,5),random(2,3,4))", "name": "g"}, "f": {"definition": "random(1..5)", "name": "f"}, "s3": {"definition": "random(1,-1)", "name": "s3"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "td": {"definition": "random(1..20)", "name": "td"}, "tc": {"definition": "s3*random(2..20)", "name": "tc"}}, "metadata": {"notes": "\n                                            \t\t    \t\t<p><strong>5/08/2012:</strong></p>\n                                            \t\t    \t\t    <p>Added more tags.</p>\n                                            \t\t    \t\t    <p>Added description.</p>\n                                            \t\t    \t\t    <p>Checked calculation. OK.</p>\n                                            \t\t    \n                                            \t\t", "description": "<p>Solve $\\displaystyle ax + b =\\frac{f}{g}( cx + d)$ for $x$.</p>\n                                    <p>A video is included in Show steps which goes through a similar example.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": [], "custom_part_types": [], "resources": []}