Simple Linear Equations, substitution
"}, "feedback": {"showactualmark": true, "feedbackmessages": [], "allowrevealanswer": true, "intro": "", "showtotalmark": true, "showanswerstate": true, "advicethreshold": 0}, "percentPass": 0, "name": "MATH6051 Homework ", "showQuestionGroupNames": false, "showstudentname": true, "duration": 0, "question_groups": [{"name": "Group", "pickQuestions": 1, "pickingStrategy": "all-ordered", "questions": [{"name": "Ida's copy of Solve a linear equation $ax+b = cx+d$", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ida Landg\u00e4rds", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2336/"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"js": "", "css": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Solve a simple linear equation algebraically. The unknown appears on both sides of the equation.
"}, "variables": {"x": {"name": "x", "templateType": "anything", "description": "", "definition": "random(2..6)", "group": "Ungrouped variables"}, "g": {"name": "g", "templateType": "anything", "description": "", "definition": "random(2..5)", "group": "Ungrouped variables"}, "dg_coprime": {"name": "dg_coprime", "templateType": "anything", "description": "", "definition": "(d-g)/gcd_hfdg", "group": "Ungrouped variables"}, "d": {"name": "d", "templateType": "anything", "description": "", "definition": "random(g+2..8)", "group": "Ungrouped variables"}, "f": {"name": "f", "templateType": "anything", "description": "", "definition": "random(2..6)", "group": "Ungrouped variables"}, "hf_coprime": {"name": "hf_coprime", "templateType": "anything", "description": "", "definition": "(h+f)/gcd_hfdg", "group": "Ungrouped variables"}, "gcd_hfdg": {"name": "gcd_hfdg", "templateType": "anything", "description": "", "definition": "gcd((h+f),(d-g))", "group": "Ungrouped variables"}, "h": {"name": "h", "templateType": "anything", "description": "", "definition": "(x*(d-g))-f", "group": "Ungrouped variables"}, "finalb": {"name": "finalb", "templateType": "anything", "description": "", "definition": "hf_coprime/dg_coprime", "group": "Ungrouped variables"}}, "variable_groups": [], "tags": ["taxonomy"], "rulesets": {}, "parts": [{"prompt": "$\\var{d}x-\\var{f}=\\var{g}x+\\var{h}$
\nWhat is the value of $x$?
\n$x = $ [[0]]
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "scripts": {}, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "finalb", "type": "numberentry", "variableReplacements": [], "showFeedbackIcon": true, "minValue": "finalb", "correctAnswerFraction": false, "marks": "2", "correctAnswerStyle": "plain", "allowFractions": false}], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "ungrouped_variables": ["d", "f", "g", "h", "x", "gcd_hfdg", "hf_coprime", "dg_coprime", "finalb"], "advice": "We are asked to solve the equation
\n\\[ \\var{d}x-\\var{f}=\\var{g}x+\\var{h} \\]
\nIn this equation, there are $x$ terms and constant terms on both sides of the equals sign.
\nTo solve this equation, we must rearrange it to get $x$ on its own.
\n\\begin{align}
\\var{d}x-\\var{f} &= \\var{g}x+\\var{h} \\\\[0.5em]
\\var{d}x-\\var{g}x &= \\var{h}+\\var{f} & \\text{Move } x \\text{ terms to the left, and constant terms to the right.}\\\\[0.5em]
\\simplify{{d-g}*x} &= {\\var{h+f}} & \\text{Collect like terms together.}\\\\[0.5em]
x &=\\frac{\\var{h+f}}{\\var{d-g}} & \\text{Divide both sides by } \\var{d-g} \\text{.} \\\\[0.5em]
x &= \\simplify{{h+f}/{d-g}}
\\end{align}
Solve a linear equation of the form $ax+b = c$, where $a$, $b$ and $c$ are integers.
\nThe answer is always an integer.
", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": [], "type": "question", "rulesets": {}, "advice": "We need to solve the equation
\n\\[ \\var{a}x+\\var{b}=\\var{c} \\]
\nTo solve this equation, we must rearrange the equation to put $x$ on its own.
\nTo do this, we should subtract $\\var{b}$ from both sides and then divide through by $\\var{a}$ to get the value for $x$.
\n\\begin{align}
\\var{a}x+\\var{b}&=\\var{c} \\\\[0.5em]
\\var{a}x&=\\var{c}-\\var{b} & \\text{Subtract } \\var{b} \\text{ from both sides} \\\\[0.5em]
\\var{a}x&=\\var{c-b} \\\\[0.5em]
x&=\\frac{\\var{c-b}}{\\var{a}} & \\text{Divide both sides by } \\var{a} \\\\[0.5em]
x&=\\simplify{{c-b}/{a}}
\\end{align}
$\\var{a}x+\\var{b}=\\var{c}$
\nWhat is the value of $x$?
\n$x = $ [[0]]
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", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["r", "x1", "n", "x2", "const", "sales"], "type": "question", "advice": "When inserting numbers into your calculator make sure you place brackets correctly.
\nAs $x = \\var{n+2}$,
\nsubstitute $\\var{n+2}$ into $\\var{x2}x^2 + \\var{x1}x + \\var{const}$.
\n\\begin{align}
\\var{x2}x^2 + \\var{x1}x + \\var{const} &= \\var{x2} (\\var{n+2})^2 + \\var{x1}(\\var{n+2}) + \\var{const} \\\\
&= \\simplify{{x2} ({n+2})^2 + {x1}({n+2}) + {const}}\\,.
\\end{align}
b)
\nAs $y = \\var{n}$,
\nsubstitute $\\var{n}$ into $\\var{n+1}y^2-\\var{x2}y$.
\n\\begin{align}
\\var{n+1}y^2-\\var{x2}y &= \\var{n+1}(\\var{n})^2-\\var{x2}(\\var{n}) \\\\
&= \\simplify{{n+1}({n})^2-{x2}({n})}\\,.
\\end{align}
c)
As we are given a temperature in degrees Celcius, $T_C = \\var{T_C}°C.$
\nSubstituting $T_C$ into $T_C = 1.8\\,T_C + 32$.
\n\\begin{align}
T_F &=1.8\\, T_C+32 \\\\
&=1.8 (\\var{T_C}) + 32 \\\\
&= \\var{dpformat(1.8 {T_C} +32, 1)}\\,°F\\,.
\\end{align}
Substitute the given values in the equations below.
", "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "allowFractions": false, "minValue": "{x2}{n+2}^2+{x1}{n+2}+{const}", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "{x2}{n+2}^2+{x1}{n+2}+{const}", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "A curve is defined by a function $y=\\simplify{{x2}x^2 + {x1}x + {const}}$.
\nWhat is the $y$ coordinate value of the point on the curve at $x=\\var{n+2}$?
\n$y =$ [[0]]
"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "allowFractions": false, "minValue": "{n+1}{n}^2-{x2}{n}", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "{n+1}{n}^2-{x2}{n}", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "{name[n]} sells luxury yachts.
\nThe predicted sales of the luxury yachts are defined by
\n\\[S=\\simplify{{n+1}y^2-{x2}y},\\]
\nwhere
$S$ is the number of sales predicted this year;
$y$ is the number of luxury yachts sold in the previous year.
{pronoun} sold {n} yachts in the previous year.
\nCalculate $S$, the number of sales predicted this year.
\n$S =$ [[0]]
"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "allowFractions": false, "minValue": "T_F", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "T_F", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "You can convert temperatures from degrees celsius to degrees fahrenheit by using the formula
\n\\[T_F=1.8\\, T_C+32,\\]
\nwhere
$T_F$ = Temperature in $°F$
$T_C$ = Temperature in $°C$.
Convert $\\var{T_C}°C$ into degrees fahrenheit.
\n$T_F =$ [[0]] $°F$
"}], "tags": ["predicted value", "substitution", "Substitution", "taxonomy"], "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"pronoun": {"description": "Defines the pronoun in the question.
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", "definition": "random(5..30#1)", "group": "Temperature conversion", "name": "T_C", "templateType": "anything"}, "name": {"description": "List of names to randomise. Can change to any name inserted
", "definition": "[\"Andrew\",\"Susan\",\"Tom\",\"Geraldine\",\"Joshua\",\"Chantel\"]", "group": "Name variables", "name": "name", "templateType": "anything"}, "n": {"description": "n is a random number between 0 and 4 that picks a name from {name} and then picks the next in the list for the other name such that there is always a male and a female in the question.
", "definition": "random(0..4#1)", "group": "Ungrouped variables", "name": "n", "templateType": "anything"}, "sales": {"description": "", "definition": "(n+1)n^2-x2*n", "group": "Ungrouped variables", "name": "sales", "templateType": "anything"}, "const": {"description": "The constant coefficient
", "definition": "random(1..100#1)", "group": "Ungrouped variables", "name": "const", "templateType": "anything"}, "T_F": {"description": "Creates a value for Temperature in fahrenheit.
", "definition": "T_C*1.8+32", "group": "Temperature conversion", "name": "T_F", "templateType": "anything"}, "r": {"description": "A random variable which will be inputted by the student.
", "definition": "random(1..50#0.1)", "group": "Ungrouped variables", "name": "r", "templateType": "anything"}, "x2": {"description": "The x^2 coefficient
", "definition": "random(1..(n+1)*n)", "group": "Ungrouped variables", "name": "x2", "templateType": "anything"}, "name2": {"description": "List of names to randomise. Can change to any name inserted
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", "definition": "random(1..50)", "group": "Ungrouped variables", "name": "x1", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}, {"name": "Q2 (Algebra equations solve for x)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "TEAME CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/591/"}], "functions": {}, "ungrouped_variables": ["num1p", "num1n", "ans11", "ans21", "numh", "num3p", "num3n", "posa", "nega", "ans41", "ans31", "posb", "neg", "ans51"], "tags": ["rebelmaths", "teame"], "preamble": {"css": "", "js": ""}, "advice": "Part 1:
\n$x$:
\n$\\frac{(\\var{numh[0]}-\\var{num1p[1]})}{(\\var{num1p[0]}-\\var{num1p[2]})} = \\var{ans11}$
\nPart 2:
\n$x$:
\n$\\frac{((\\var{num1p[7]} \\times \\var{num1p[6]})-(\\var{num1p[4]} \\times \\var{num1n[2]})}{((\\var{num1p[1]} \\times \\var{num1p[5]})-(\\var{num1p[7]} \\times \\var{num1n[7]})} = \\var{ans21}$
\nPart 3:
\n$x$:
\n$\\frac{((\\var{num3p[4]} \\times \\var{num3p[5]})+\\var{num3p[6]}+(\\var{num3n[4]} \\times \\var{num3n[5]})-\\var{num3p[0]}-(\\var{num3p[1]} \\times \\var{num3n[0]})-(\\var{num3n[1]} \\times \\var{num3n[2]}))} {((\\var{num3p[1]} \\times \\var{num3p[2]})+(\\var{num3n[1]} \\times \\var{num3p[3]})-(\\var{num3p[4]} \\times \\var{num3n[3]})-(\\var{num3n[4]} \\times \\var{num3p[7]}))} = \\var{ans31}$
\nPart 4:
\n$x$:
\n$\\frac{(\\var{nega[3]}-\\frac{\\var{nega[1]}}{\\var{posa[1]}}+\\frac{\\var{posa[5]}}{\\var{posa[3]}}-\\frac{\\var{nega[2]}}{\\var{posa[6]}})} {(\\frac{\\var{posa[2]}}{\\var{posa[1]}}-\\frac{\\var{posa[4]}}{\\var{posa[3]}}+\\frac{\\var{posa[7]}}{\\var{posa[6]}})} = \\var{ans41}$
\nPart 5:
\n$x$:
\n$\\frac{((\\var{posb[3]} \\times \\var{nega[4]}) - (\\var{posb[1]} \\times \\var{posb[5]}))}{((\\var{posb[1]} \\times \\var{posb[4]}) - (\\var{posb[3]} \\times \\var{posb[2]}))} = \\var{ans51}$
", "rulesets": {"std": ["all", "basic", "simplifyFractions"], "ruleset1": []}, "parts": [{"prompt": "$\\var{num1p[0]}x + \\var{num1p[1]} = \\var{num1p[2]}x + \\var{numh[0]}$
\n$x =$ [[0]]
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\n$x =$ [[0]]
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "({ans21}+0.5)", "strictPrecision": false, "minValue": "({ans21}-0.5)", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": "0", "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "$\\var{num3p[0]} + \\var{num3p[1]}(\\var{num3p[2]}x \\var{num3n[0]})\\var{num3n[1]}(\\var{num3p[3]}x \\var{num3n[2]}) = \\var{num3p[4]}(\\var{num3p[5]} \\var{num3n[3]}x) + \\var{num3p[6]} \\var{num3n[4]}(\\var{num3p[7]}x \\var{num3n[5]})$
\n$x =$ [[0]]
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "{ans31}+0.5", "strictPrecision": false, "minValue": "{ans31}-0.5", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "$\\frac{1}{\\var{posa[1]}}(\\var{posa[2]}x \\var{nega[1]})-\\frac{1}{\\var{posa[3]}}(\\var{posa[4]}x +\\var{posa[5]})+\\frac{1}{\\var{posa[6]}}(\\var{posa[7]}x \\var{nega[2]}) = \\var{nega[3]}$
\n$x =$ [[0]]
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "Get a common denominator like in question 1. The multiply both sides of the equation by that common denominator to get rid of the fractions.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "{ans41}+0.5", "strictPrecision": false, "minValue": "{ans41}-0.5", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "$\\frac{\\var{posb[1]}}{\\var{posb[2]}x\\var{neg[4]}}= \\frac{\\var{posb[3]}}{\\var{posb[4]}x+\\var{posb[5]}}$
\n$x =$ [[0]]
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "{ans51}+0.5", "strictPrecision": false, "minValue": "{ans51}-0.5", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "Solve for $x$ in the following, to 2 decimal places:
\nWatch the video below for help with the questions:
\n", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"num3n": {"definition": "shuffle(-10..-2)[0..9]", "templateType": "anything", "group": "Ungrouped variables", "name": "num3n", "description": ""}, "num1n": {"definition": "shuffle(-9..-2)[0..9]", "templateType": "anything", "group": "Ungrouped variables", "name": "num1n", "description": ""}, "ans41": {"definition": "(nega[3]-(nega[1]/posa[1])+(posa[5]/posa[3])-(nega[2]/posa[6]))/((posa[2]/posa[1])-(posa[4]/posa[3])+(posa[7]/posa[6]))\n", "templateType": "anything", "group": "Ungrouped variables", "name": "ans41", "description": ""}, "posb": {"definition": "shuffle(2..10)[0..9]", "templateType": "anything", "group": "Ungrouped variables", "name": "posb", "description": ""}, "posa": {"definition": "shuffle(2..10)[0..9]", "templateType": "anything", "group": "Ungrouped variables", "name": "posa", "description": ""}, "ans11": {"definition": "(numh[0]-num1p[1])/(num1p[0]-num1p[2])", "templateType": "anything", "group": "Ungrouped variables", "name": "ans11", "description": ""}, "num1p": {"definition": "shuffle(2..10)[0..9]", "templateType": "anything", "group": "Ungrouped variables", "name": "num1p", "description": ""}, "neg": {"definition": "shuffle(-9..-2)[0..9]", "templateType": "anything", "group": "Ungrouped variables", "name": "neg", "description": ""}, "ans31": {"definition": "((num3p[4] * num3p[5])+num3p[6]+(num3n[4] *num3n[5])-num3p[0]-(num3p[1] * num3n[0])-(num3n[1] * num3n[2])) / ((num3p[1] * num3p[2])+(num3n[1] * num3p[3])-(num3p[4] * num3n[3])-(num3n[4] * num3p[7])) ", "templateType": "anything", "group": "Ungrouped variables", "name": "ans31", "description": ""}, "nega": {"definition": "shuffle(-9..-2)[0..9]", "templateType": "anything", "group": "Ungrouped variables", "name": "nega", "description": ""}, "num3p": {"definition": "shuffle(2..10)[0..9]", "templateType": "anything", "group": "Ungrouped variables", "name": "num3p", "description": ""}, "numh": {"definition": "shuffle(20..40)[0..15]", "templateType": "anything", "group": "Ungrouped variables", "name": "numh", "description": ""}, "ans21": {"definition": "((num1p[7] * num1p[6])-(num1p[4] * num1p[5]))/((num1p[4] * num1n[2])-(num1p[7] * num1n[7])) ", "templateType": "anything", "group": "Ungrouped variables", "name": "ans21", "description": ""}, "ans51": {"definition": "((posb[3] * neg[4]) - (posb[1] * posb[5]))/((posb[1] * posb[4]) - (posb[3] * posb[2]))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans51", "description": ""}}, "metadata": {"description": "Simple Linear Equations involving basic transposition
\nrebelmaths
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "navigation": {"preventleave": true, "showresultspage": "oncompletion", "reverse": true, "onleave": {"action": "none", "message": ""}, "showfrontpage": true, "allowregen": true, "browse": true}, "type": "exam", "contributors": [{"name": "Deirdre Casey", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/681/"}], "extensions": ["geogebra"], "custom_part_types": [], "resources": []}