// Numbas version: exam_results_page_options {"name": "Solve an equation with reciprocals", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": [], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "statement": "\n
Solve the following equation for $x$.
\nInput your answer as a fraction or an integer as appropriate and not as a decimal.
\n ", "name": "Solve an equation with reciprocals", "parts": [{"useCustomName": false, "prompt": "\n\\[\\simplify{{s} / ({a} * x + {b}) = {t} / ({c} * x + {d})}\\]
\n$x=\\;$ [[0]]
\nIf you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.
\n \n \n ", "sortAnswers": false, "steps": [{"useCustomName": false, "prompt": "\nRearrange the equation by cross-multiplying to get:
\\[\\simplify{{s}*({c} * x + {d}) = {t} *({a} * x + {b})}\\]
Multiply out to get \\[\\simplify{{s*c}*x+{s*d}={t*a}*x+{t*b}}.\\] Now solve this linear equation.
Input as a fraction or an integer, not as a decimal.
"}, "showPreview": true, "marks": 2, "failureRate": 1, "checkVariableNames": false, "showFeedbackIcon": true, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "customName": "", "checkingAccuracy": 0.0001, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": ""}], "variableReplacementStrategy": "originalfirst"}], "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "preamble": {"js": "", "css": ""}, "variables": {"d": {"definition": "random(-9..9 except [0,t])", "description": "", "templateType": "anything", "name": "d", "group": "Ungrouped variables"}, "b": {"definition": "random(-9..9 except [0,s])", "description": "", "templateType": "anything", "name": "b", "group": "Ungrouped variables"}, "t": {"definition": "random(2..8 except s)", "description": "", "templateType": "anything", "name": "t", "group": "Ungrouped variables"}, "c": {"definition": "random(2..9 except [s,abs(d),a*t/s])", "description": "", "templateType": "anything", "name": "c", "group": "Ungrouped variables"}, "a": {"definition": "random(2..9 except [s,abs(b)])", "description": "", "templateType": "anything", "name": "a", "group": "Ungrouped variables"}, "an2": {"definition": "s*c-a*t", "description": "", "templateType": "anything", "name": "an2", "group": "Ungrouped variables"}, "s": {"definition": "random(2..8)", "description": "", "templateType": "anything", "name": "s", "group": "Ungrouped variables"}, "an1": {"definition": "b*t-s*d", "description": "", "templateType": "anything", "name": "an1", "group": "Ungrouped variables"}}, "metadata": {"description": "Solve for $x$: $\\displaystyle \\frac{s}{ax+b} = \\frac{t}{cx+d}$
", "licence": "Creative Commons Attribution 4.0 International"}, "extensions": [], "ungrouped_variables": ["a", "c", "b", "d", "s", "t", "an2", "an1"], "advice": "Rearrange the equation by cross-multiplying to get:
\\[\\simplify{{s}*({c} * x + {d}) = {t} *({a} * x + {b})}\\]
Multiply out to get \\[\\simplify{{s*c}*x+{s*d}={t*a}*x+{t*b}}.\\] Now this is a linear equation which is solved in the following steps: \\[\\simplify{{s*c-t*a}*x={t*b-s*d}}\\] and then \\[\\simplify{x={t*b-s*d}/{s*c-t*a}}.\\]