// Numbas version: exam_results_page_options {"name": "Solving linear equations: collecting like terms and solving", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"css": "", "js": ""}, "variables": {"add": {"templateType": "anything", "definition": "primes[2]", "name": "add", "group": "e", "description": ""}, "r1": {"templateType": "anything", "definition": "right*denom1", "name": "r1", "group": "e", "description": ""}, "q": {"templateType": "anything", "definition": "random(1..12)", "name": "q", "group": "Ungrouped variables", "description": ""}, "eans": {"templateType": "anything", "definition": "denom2*(right*denom1-add)/(denom2+denom1)", "name": "eans", "group": "e", "description": ""}, "denom1": {"templateType": "anything", "definition": "primes[0]", "name": "denom1", "group": "e", "description": ""}, "right": {"templateType": "anything", "definition": "random(-5..5 except 0)", "name": "right", "group": "e", "description": ""}, "c": {"templateType": "anything", "definition": "a+random(1..12)", "name": "c", "group": "Ungrouped variables", "description": ""}, "g": {"templateType": "anything", "definition": "random(-12..-1)", "name": "g", "group": "Ungrouped variables", "description": ""}, "r12": {"templateType": "anything", "definition": "r1*denom2", "name": "r12", "group": "e", "description": ""}, "a": {"templateType": "anything", "definition": "random(1..12)", "name": "a", "group": "Ungrouped variables", "description": ""}, "k": {"templateType": "anything", "definition": "random(1..12)", "name": "k", "group": "Ungrouped variables", "description": ""}, "l": {"templateType": "anything", "definition": "random(2..12)", "name": "l", "group": "Ungrouped variables", "description": ""}, "sumdeno": {"templateType": "anything", "definition": "denom1+denom2", "name": "sumdeno", "group": "e", "description": ""}, "ans1": {"templateType": "anything", "definition": "-b/(a-c)", "name": "ans1", "group": "Ungrouped variables", "description": ""}, "a2": {"templateType": "anything", "definition": "add*denom2", "name": "a2", "group": "e", "description": ""}, "j": {"templateType": "anything", "definition": "random(1..12 except h)", "name": "j", "group": "Ungrouped variables", "description": ""}, "p": {"templateType": "anything", "definition": "random(1..3)", "name": "p", "group": "Ungrouped variables", "description": ""}, "f": {"templateType": "anything", "definition": "random(2..12)", "name": "f", "group": "Ungrouped variables", "description": ""}, "h": {"templateType": "anything", "definition": "random(1..12)", "name": "h", "group": "Ungrouped variables", "description": ""}, "n": {"templateType": "anything", "definition": "random(1..12)", "name": "n", "group": "Ungrouped variables", "description": ""}, "ans4": {"templateType": "anything", "definition": "{q+n*l}/{l*m-p}", "name": "ans4", "group": "Ungrouped variables", "description": ""}, "primes": {"templateType": "anything", "definition": "shuffle([2,3,5,7,11])", "name": "primes", "group": "e", "description": ""}, "denom2": {"templateType": "anything", "definition": "primes[1]", "name": "denom2", "group": "e", "description": ""}, "m": {"templateType": "anything", "definition": "random(2..12)", "name": "m", "group": "Ungrouped variables", "description": ""}, "ans3": {"templateType": "anything", "definition": "(k+j)*h", "name": "ans3", "group": "Ungrouped variables", "description": ""}, "top": {"templateType": "anything", "definition": "r12-a2", "name": "top", "group": "e", "description": ""}, "ans2": {"templateType": "anything", "definition": "{g*f}/{d-g}", "name": "ans2", "group": "Ungrouped variables", "description": ""}, "d": {"templateType": "anything", "definition": "random(1..12)", "name": "d", "group": "Ungrouped variables", "description": ""}, "b": {"templateType": "anything", "definition": "random(1..12)", "name": "b", "group": "Ungrouped variables", "description": ""}}, "advice": "", "functions": {}, "rulesets": {}, "parts": [{"extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "sortAnswers": false, "stepsPenalty": "1", "variableReplacementStrategy": "originalfirst", "prompt": "
We can solve $\\var{a}x+\\var{b}=\\var{c}x$ by collecting like terms and rearranging for $x$. This gives $x=$ [[0]].
", "customMarkingAlgorithm": "", "scripts": {}, "type": "gapfill", "variableReplacements": [], "steps": [{"marks": 0, "extendBaseMarkingAlgorithm": true, "scripts": {}, "variableReplacements": [], "prompt": "Given $\\var{a}x+\\var{b}=\\var{c}x$, we can subtract $\\var{a}x$ from both sides to collect like terms, and then divide both sides by the coefficient of $x$ to get $x$ by itself.
\n\n$\\var{a}x+\\var{b}$ | \n$=$ | \n$\\var{c}x$ | \n
\n | \n | \n |
$\\var{a}x+\\var{b}-\\var{a}x$ | \n$=$ | \n$\\var{c}x-\\var{a}x$ | \n
\n | \n | \n |
$\\var{b}$ | \n$=$ | \n$\\var{c-a}x$ | \n
\n | \n | \n |
$\\displaystyle{\\frac{\\var{b}}{\\var{c-a}}}$ | \n$=$ | \n$\\displaystyle{\\frac{\\var{c-a}x}{\\var{c-a}}}$ | \n
\n | \n | \n |
$\\displaystyle{\\simplify{{b}/{c-a}}}$ | \n$=$ | \n$x$ | \n
\n | \n | \n |
$x$ | \n$=$ | \n$\\displaystyle{\\simplify{{b}/{c-a}}}$ | \n
There is often more than one way to solve an equation, one strategy used above in the first step was to get all the $x$'s one the side with the most $x$'s, that way you end up with a postive number of $x$'s. This is not necessary, we could have put the all the $x$'s on the left hand side but notice in this question we then would have had to move the $\\var{b}$ on to the right hand side, so it would have required more work, but nevertheless that method would result in the same result for $x$.
", "type": "information", "showFeedbackIcon": true, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "unitTests": []}], "gaps": [{"maxValue": "{-b}/{a-c}", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "correctAnswerFraction": true, "scripts": {}, "variableReplacements": [], "type": "numberentry", "allowFractions": true, "showCorrectAnswer": true, "minValue": "{-b}/{a-c}", "unitTests": []}], "showCorrectAnswer": true, "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "sortAnswers": false, "stepsPenalty": "1", "variableReplacementStrategy": "originalfirst", "prompt": "Solve $\\var{l}(\\var{m}w-\\var{n})=\\var{p}w+\\var{q}$ for $w$.
\n$w=$ [[0]]
", "customMarkingAlgorithm": "", "scripts": {}, "type": "gapfill", "variableReplacements": [], "steps": [{"marks": 0, "extendBaseMarkingAlgorithm": true, "scripts": {}, "variableReplacements": [], "prompt": "Given $\\var{l}(\\var{m}w-\\var{n})=\\var{p}w+\\var{q}$, we can expand the brackets, get all the $w$'s on the left hand side and all the numbers on the right hand side, and then divide both sides by the coefficient of $w$ to get $w$ by itself.
\n\n$\\var{l}(\\var{m}w-\\var{n})$ | \n$=$ | \n$\\var{p}w+\\var{q}$ | \n
\n | \n | \n |
$\\var{l*m}w-\\var{n*l}$ | \n$=$ | \n$\\var{p}w+\\var{q}$ | \n
\n | \n | \n |
$\\var{l*m}w-\\var{n*l}-\\var{p}w$ | \n$=$ | \n$\\var{p}w+\\var{q}-\\var{p}w$ | \n
\n | \n | \n |
$\\var{l*m-p}w-\\var{n*l}$ | \n$=$ | \n$\\var{q}$ | \n
\n | \n | \n |
$\\var{l*m-p}w-\\var{n*l}+\\var{n*l}$ | \n$=$ | \n$\\var{q}+\\var{n*l}$ | \n
\n | \n | \n |
$\\var{l*m-p}w$ | \n$=$ | \n$\\var{q+n*l}$ | \n
\n | \n | \n |
$\\displaystyle{\\frac{\\var{l*m-p}w}{\\var{l*m-p}}}$ | \n$=$ | \n$\\displaystyle{\\frac{\\var{q+n*l}}{\\var{l*m-p}}}$ | \n
\n | \n | \n |
$w$ | \n$=$ | \n$\\displaystyle{\\simplify{{q+n*l}/{l*m-p}}}$ | \n
Given $\\displaystyle{\\frac{\\var{d}y}{y-\\var{f}}}=\\var{g}$, $y=$ [[0]].
", "customMarkingAlgorithm": "", "scripts": {}, "type": "gapfill", "variableReplacements": [], "steps": [{"marks": 0, "extendBaseMarkingAlgorithm": true, "scripts": {}, "variableReplacements": [], "prompt": "Given $\\displaystyle{\\frac{\\var{d}y}{y-\\var{f}}}=\\var{g}$, we can multiply both sides by $(y-\\var{f})$ to get rid of the fraction, get all the $y$'s on one side and the numbers on the other side, and then divide both sides by the coefficient of $y$ to get $y$ by itself.
\n\n$\\displaystyle{\\frac{\\var{d}y}{y-\\var{f}}}$ | \n$=$ | \n$\\var{g}$ | \n
\n | \n | \n |
$\\displaystyle{\\frac{\\var{d}y}{y-\\var{f}}}\\times(y-\\var{f})$ | \n$=$ | \n$\\var{g}\\times (y-\\var{f})$ | \n
\n | \n | \n |
$\\var{d}y$ | \n$=$ | \n$\\var{g}y+\\var{-g*f}$ | \n
\n | \n | \n |
$\\var{d}y+\\var{-g}y$ | \n$=$ | \n$\\var{g}y+\\var{-g*f}+\\var{-g}y$ | \n
\n | \n | \n |
$\\var{d-g}y$ | \n$=$ | \n$\\var{-g*f}$ | \n
\n | \n | \n |
$\\displaystyle{\\frac{\\var{d-g}y}{\\var{d-g}}}$ | \n$=$ | \n$\\displaystyle{\\frac{\\var{-g*f}}{\\var{d-g}}}$ | \n
\n | \n | \n |
$y$ | \n$=$ | \n$\\displaystyle{\\simplify{{-g*f}/{d-g}}}$ | \n
Solve $\\displaystyle{\\frac{z+\\var{h}}{z+\\var{j}}}=\\var{k}$ for $z$.
\n$z=$ [[0]]
", "customMarkingAlgorithm": "", "scripts": {}, "type": "gapfill", "variableReplacements": [], "steps": [{"marks": 0, "extendBaseMarkingAlgorithm": true, "scripts": {}, "variableReplacements": [], "prompt": "Given $\\displaystyle{\\frac{z+\\var{h}}{z+\\var{j}}}=\\var{k}$, we can multiply both sides by $(z+\\var{j})$ to get rid of the fraction, get all the $z$'s on one side and the numbers on the other side, and then divide both sides by the coefficient of $z$ to get $z$ by itself.
\n\n$\\displaystyle{\\frac{z+\\var{h}}{z+\\var{j}}}$ | \n$=$ | \n$\\var{k}$ | \n
\n | \n | \n |
$\\displaystyle{\\frac{z+\\var{h}}{z+\\var{j}}}\\times(z+\\var{j})$ | \n$=$ | \n$\\var{k}\\times (z+\\var{j})$ | \n
\n | \n | \n |
$z+\\var{h}$ | \n$=$ | \n$\\var{k}z+\\var{k*j}$ | \n
\n | \n | \n |
$z+\\var{h}-\\var{k}z$ | \n$=$ | \n$\\var{k}z+\\var{k*j}-\\var{k}z$ | \n
\n | \n | \n |
$\\var{1-k}z+\\var{h}$ | \n$=$ | \n$\\var{k*j}$ | \n
\n | \n | \n |
$\\var{1-k}z+\\var{h}-\\var{h}$ | \n$=$ | \n$\\var{k*j}-\\var{h}$ | \n
\n | \n | \n |
$\\var{1-k}z$ | \n$=$ | \n$\\var{k*j-h}$ | \n
\n | \n | \n |
$\\displaystyle{\\frac{\\var{1-k}z}{\\var{1-k}}}$ | \n$=$ | \n$\\displaystyle{\\frac{\\var{k*j-h}}{\\var{1-k}}}$ | \n
\n | \n | \n |
$z$ | \n$=$ | \n$\\displaystyle{\\simplify{({k*j-h})/({1-k})}}$ | \n
Solve $\\displaystyle{\\frac{x+\\var{add}}{\\var{denom1}}+\\frac{x}{\\var{denom2}}=\\var{right}}$.
\n$x=$ [[0]]
", "customMarkingAlgorithm": "", "scripts": {}, "type": "gapfill", "variableReplacements": [], "steps": [{"marks": 0, "extendBaseMarkingAlgorithm": true, "scripts": {}, "variableReplacements": [], "prompt": "Given $\\displaystyle{\\frac{x+\\var{add}}{\\var{denom1}}+\\frac{x}{\\var{denom2}}=\\var{right}}$, we can multiply both sides by $\\var{denom1}$ and by $\\var{denom2}$ to get rid of the fractions, get all the $x$'s on one side and the numbers on the other side, and then divide both sides by the coefficient of $x$ to get $x$ by itself.
\n\n$\\displaystyle{\\frac{x+\\var{add}}{\\var{denom1}}+\\frac{x}{\\var{denom2}}}$ | \n$=$ | \n$\\var{right}$ | \n\n |
\n | \n | \n | \n |
$\\displaystyle{\\left(\\frac{x+\\var{add}}{\\var{denom1}}\\right)\\times\\var{denom1}+\\left(\\frac{x}{\\var{denom2}}\\right)\\times\\var{denom1}}$ | \n$=$ | \n$\\var{right}\\times \\var{denom1}$ | \n(multiply all terms by $\\var{denom1}$) | \n
\n | \n | \n | \n |
$\\displaystyle{x+\\var{add}+\\frac{\\var{denom1}x}{\\var{denom2}}}$ | \n$=$ | \n$\\var{r1}$ | \n\n |
\n | \n | \n | \n |
$\\displaystyle{(x+\\var{add})\\times\\var{denom2}+\\left(\\frac{\\var{denom1}x}{\\var{denom2}}\\right)\\times\\var{denom2}}$ | \n$=$ | \n$\\var{r1}\\times\\var{denom2}$ | \n(multiply all terms by $\\var{denom2}$) | \n
\n | \n | \n | \n |
$\\displaystyle{\\var{denom2}x+\\var{a2}+\\var{denom1}x}$ | \n$=$ | \n$\\var{r12}$ | \n\n |
\n | \n | \n | \n |
$\\var{sumdeno}x+\\var{a2}$ | \n$=$ | \n$\\var{r12}$ | \n(collect like terms) | \n
\n | \n | \n | \n |
$\\var{sumdeno}x$ | \n$=$ | \n$\\var{r12}-\\var{a2}$ | \n(collect like terms) | \n
\n | \n | \n | \n |
$\\var{sumdeno}x$ | \n$=$ | \n$\\var{top}$ | \n\n |
\n | \n | \n | \n |
$x$ | \n$=$ | \n$\\displaystyle{\\simplify{{top}/({sumdeno})}}$ | \n(divide by the coefficient of $x$) | \n