Input your answer as a fraction and not a decimal.
"}, "checkingType": "absdiff", "customMarkingAlgorithm": "", "unitTests": []}, {"variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "expectedVariableNames": [], "showPreview": true, "vsetRange": [0, 1], "marks": 1, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "answer": "{-c*a1+c1*a}/{a*b1-a1*b}", "checkVariableNames": false, "type": "jme", "checkingAccuracy": 0.001, "failureRate": 1, "scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "answerSimplification": "Std", "notallowed": {"strings": ["."], "partialCredit": 0, "showStrings": false, "message": "Input your answer as a fraction and not as a decimal.
"}, "checkingType": "absdiff", "customMarkingAlgorithm": "", "unitTests": []}], "marks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "prompt": "$x=$ [[0]]
\n$y=$ [[1]]
\nInput your answers as fractions and not as decimals.
", "scripts": {}, "showFeedbackIcon": true, "sortAnswers": false, "showCorrectAnswer": true, "type": "gapfill", "customMarkingAlgorithm": "", "unitTests": []}], "advice": "Multiply the first equation by $\\var{b1}$ and the second equation by $\\var{b}$ so they both have the same $y$ coefficient:
\n\\begin{align}
\\simplify{{a*b1}x+{b*b1}y} &= \\var{c*b1} \\\\
\\simplify{{a1*b}x+{b1*b}y} &= \\var{c1*b}
\\end{align}
Next, subtract the second equation from the first to get
\n\\[ \\simplify[std]{{a*b1-a1*b}x} = \\var{c*b1-c1*b} \\]
\nSo $x = \\simplify[std]{{(c*b1-c1*b)/(a*b1-a1*b)}}$.
\nSubstitute this value of $x$ into the first equation and rearrange to obtain $y$:
\n\\begin{align}
\\simplify[std]{{a}*{(c*b1-c1*b)/(a*b1-a1*b)} + {b}y} &= \\var{c} \\\\
\\simplify[std]{{b}y} &= \\simplify[std]{{c}-{a*(c*b1-c1*b)/(a*b1-a1*b)}} \\\\
y &= \\simplify[std]{{(c-a*(c*b1-c1*b)/(a*b1-a1*b))/b}}
\\end{align}
Solve:
\n\\[\\begin{eqnarray*} \\simplify{{a}x+{b}y}&=&\\var{c}\\\\\\\\\\simplify{{a1}x+{b1}y}&=&\\var{c1}\\end{eqnarray*}\\]
", "preamble": {"js": "", "css": ""}, "variable_groups": [], "metadata": {"description": "Shows how to define variables to stop degenerate examples.
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", "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "marks": 0, "scripts": {}, "type": "gapfill", "gaps": [{"vsetRangePoints": 5, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "marks": 1, "scripts": {}, "expectedVariableNames": [], "answer": "{c*b1-b*c1}/{b1*a-a1*b}", "type": "jme", "showPreview": true, "notallowed": {"partialCredit": 0, "message": "Input as a fraction or an integer not as a decimal
", "showStrings": false, "strings": ["."]}, "vsetRange": [0, 1], "showCorrectAnswer": true, "failureRate": 1, "answerSimplification": "std", "showFeedbackIcon": true, "unitTests": [], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "variableReplacements": [], "checkVariableNames": false}, {"vsetRangePoints": 5, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "marks": 1, "scripts": {}, "expectedVariableNames": [], "answer": "{c*a1-a*c1}/{b*a1-a*b1}", "type": "jme", "showPreview": true, "notallowed": {"partialCredit": 0, "message": "Input as a fraction or an integer not as a decimal
", "showStrings": false, "strings": ["."]}, "vsetRange": [0, 1], "showCorrectAnswer": true, "failureRate": 1, "answerSimplification": "std", "showFeedbackIcon": true, "unitTests": [], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "variableReplacements": [], "checkVariableNames": false}], "showCorrectAnswer": true, "showFeedbackIcon": true, "unitTests": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "\\[ \\begin{eqnarray} \\simplify[std]{{a}x+{b}y}&=&\\var{c}\\\\ \\simplify[std]{{a1}x+{b1}y}&=&\\var{c1} \\end{eqnarray} \\]
\n$x=\\phantom{{}}$[[0]], $y=\\phantom{{}}$[[1]]
\nInput your answers as fractions or integers, not as decimals.
\n"}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$ and $y$: \\[ \\begin{eqnarray} a_1x+b_1y&=&c_1\\\\ a_2x+b_2y&=&c_2 \\end{eqnarray} \\]
\nThe included video describes a more direct method of solving when, for example, one of the equations gives a variable directly in terms of the other variable.
"}, "variablesTest": {"condition": "{a}<>{a1} and {b}<>{b1}", "maxRuns": 100}, "tags": [], "ungrouped_variables": ["a", "c", "b", "that", "this", "fromorto", "s1", "s6", "a1", "aort", "a2", "b1", "b2", "sc", "sb", "sa", "sc1", "c1"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "functions": {}, "advice": "\\[ \\begin{eqnarray} \\simplify[std]{{a}x+{b}y}&=&\\var{c}&\\mbox{ ........(1)}\\\\ \\simplify[std]{{a1}x+{b1}y}&=&\\var{c1}&\\mbox{ ........(2)} \\end{eqnarray} \\]
To get a solution for $x$ multiply equation (1) by {this} and equation (2) by {that}
This gives:
\\[ \\begin{eqnarray} \\simplify[std]{{a*this}x+{b*this}y}&=&\\var{this*c}&\\mbox{ ........(3)}\\\\ \\simplify[std]{{a1*that}x+{b1*that}y}&=&\\var{that*c1}&\\mbox{ ........(4)} \\end{eqnarray} \\]
Now {aort} (4) {fromorto} equation (3) to get
\\[\\simplify[std]{({a*this}+{s6*a1*that})x={this*c}+{s6*that*c1}}\\]
And so we get the solution for $x$:
\\[x = \\simplify{{c*b1-b*c1}/{b1*a-a1*b}}\\]
Substituting this value into either of the equations (1) and (2), then solving for $y$ gives:
\\[y = \\simplify{{c*a1-a*c1}/{b*a1-a*b1}}\\]
You can check that these solutions are correct by seeing if they satisfy both equations (1) and (2) by substituting these values into the equations.
(a)
\nLet $L$ be the wage for 1 labourer, and let $T$ be the wage for 1 tradesman
\nThen we can set up two simultaneous equations as follows:
\n\n$\\var{num41}T+\\var{num42}L=\\var{wages41}$ (Equation 1)
\n$T + \\var{num43}L=\\var{wages42}$ (Equation 2)
\n\nMultiplying equation 2 by $\\var{num41}$ gives:
\n$\\var{num41}T+\\var{num41*num43}L=\\var{num41*wages42}$ (Equation 3)
\n\nSubtracting equation 1 from equation 3 eliminates $T$ and gives:
\n$\\var{num41*num43}L-\\var{num42}L=\\var{num41*wages42}-\\var{wages41}$
\nwhich we can solve for $L$ as follows:
\n$\\var{-num42+num41*num43}L=\\var{-wages41+num41*wages42}$
\n$L=\\var{precround((wages41-num41*wages42)/(num42-num41*num43),2)}$
\n\nWe can now find $T$ by substituting this value of $L$ into any of Equations 1-3.
\ne.g. Substituting into Equation 2 gives:
\n$T + \\var{num43}\\times \\var{precround((wages41-num41*wages42)/(num42-num41*num43),2)} = \\var{wages42}$
\n$T= \\var{precround(wages42-num43*(wages41-num41*wages42)/(num42-num41*num43),2)} $
\n\n\n(b)
\n$\\var{num11}$ tradesmen and $\\var{num12}$ labourers earn $€\\var{wages1}$ between them to do a job. If a tradesman earns $€\\var{num13}$ more than a labourer, calculate the earnings for a tradesman and a labourer.
\n\nWe can set up two simultaneous equations as follows:
\n$\\var{num11}T+\\var{num12}L=\\var{wages1}$ (Equation 1)
\n$T=L+\\var{num13}$ (Equation 2)
\n\nWe can replace $T$ in Equation 1 with $L+\\var{num13}$ to obtain:
\n$\\var{num11}(L+\\var{num13})+\\var{num12}L=\\var{wages1}$
\nHence:
\n$\\var{num11+num12}L+\\var{num11*num13}=\\var{wages1}$
\n$\\var{num11+num12}L=\\var{wages1-num11*num13}$
\n$L=\\var{precround((wages1-num11*num13)/(num11+num12),2)}$
\n\nUsing Equation 2 we also obtain:
\n$T=\\var{precround((wages1-num11*num13)/(num11+num12),2)}+\\var{num13}=\\var{precround(num13+((wages1-num11*num13)/(num11+num12)),2)}$
\n\n\n(c)
\n$\\var{num21}$ workmen on a building site earn a total of $€\\var{wages2}$ between them per week. Labourers are paid $€\\var{num22}$ per week and Tradesmen are paid $€\\var{num23}$ per week. How many of each is employed?
\nWe can set up two simultaneous equations as follows:
\n$\\var{num23}T+\\var{num22}L = \\var{wages2}$ (Equation 1)
\n$T+L = \\var{num21}$ (Equation 2)
\n\nMultiplying equation 2 by $\\var{num23}$ gives:
\n$\\var{num23}T+\\var{num23}L=\\var{num21*num23}$ (Equation 3)
\n\nSubtracting equation 1 from equation 3 eliminates $T$ and gives:
\n$\\var{num23}L-\\var{num22}L=\\var{num21*num23}-\\var{wages2}$
\nwhich we can solve for $L$ as follows:
\n$\\var{num23-num22}L=\\var{-wages2+num21*num23}$
\n$L=\\var{precround((-wages2+num21*num23)/(num23-num22),0)}$
\n\nThen substituting this value of $L$ into Equation 2 gives:
\n$T+\\var{precround((-wages2+num21*num23)/(num23-num22),0)} = \\var{num21}$
\n$T = \\var{num21-precround((-wages2+num21*num23)/(num23-num22),0)}$
\n\n\n\n\n", "functions": {}, "statement": "Calculate the following, to the nearest cent!
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\nTradesmen: €[[0]]
\nLabourer: €[[1]]
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\nTradesmen: €[[0]]
\nLabourer: €[[1]]
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\n[[0]] Labourers
\n[[1]] Tradesmen
", "unitTests": [], "marks": 0, "scripts": {}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacements": []}], "ungrouped_variables": ["num11", "num12", "wages1", "num13", "ans11", "ans12", "ans21", "ans22", "num21", "num22", "num23", "wages2", "num31", "num32", "wages3", "num33", "ans31", "ans32", "num41", "num42", "wages41", "num43", "wages42", "ans41", "ans42"], "variable_groups": [], "variables": {"wages41": {"name": "wages41", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(9000..11000)"}, "ans12": {"name": "ans12", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "(wages1-(num11*num13))/(num11+num12)"}, "num13": {"name": "num13", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(71..89 except[80])"}, "num31": {"name": "num31", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(4..11 except [num11,num21])"}, "num21": {"name": "num21", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "ans21+ans22"}, "ans22": {"name": "ans22", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(3..12 except ans21)"}, "wages42": {"name": "wages42", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(2405..2795#5)"}, "wages3": {"name": "wages3", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(7005..8995#5)"}, "num43": {"name": "num43", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(3..4)"}, "num22": {"name": "num22", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(500..600#5)"}, "num41": {"name": "num41", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(5..7)"}, "num12": {"name": "num12", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(4..7 except num11)"}, "ans21": {"name": "ans21", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(5..9)"}, "ans11": {"name": "ans11", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "ans12+num13"}, "ans31": {"name": "ans31", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "ans32+num33"}, "ans42": {"name": "ans42", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "wages42-(num43*ans41)"}, "num32": {"name": "num32", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(4..11 except [num11,num21,num31])"}, "ans32": {"name": "ans32", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "(wages3-(num31*num33))/(num31+num32)"}, "wages2": {"name": "wages2", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "(ans21*num22)+(ans22*num23)"}, "num33": {"name": "num33", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(66..84 except[70,75,80])"}, "wages1": {"name": "wages1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(8000..9000)"}, "num42": {"name": "num42", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(7..9 except num41)"}, "num23": {"name": "num23", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(num22+140..800#5)"}, "ans41": {"name": "ans41", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "(wages41-(wages42*num41))/(num42 -(num41*num43))"}, "num11": {"name": "num11", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(7..9)"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Calculating wages using algebraic equations
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"}, "rulesets": {}, "type": "question"}, {"name": "Simon's copy of Q4 (Calculating area using algebraic equations)", "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/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}], "functions": {}, "variable_groups": [], "ungrouped_variables": ["lent1", "red1", "area1", "ans11", "ans12", "per2", "ratio2", "ans21", "ans22", "ans23", "lent3", "per3", "ans31", "ans32", "ratio4", "per4", "ans41", "ans42", "ans43"], "statement": "Solve the following correct to 3 decimal place:
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\nFind the width, length and area of a rectangular room which has a perimeter of $\\var{per2} m$, if its length is $\\var{ratio2}$ times its width.
\n\nLet the width be $w$ and the length be $l$. Then from the information in the question we obtain the following:
\n$l=\\var{ratio2}w$ (Equation 1)
\n$2l+2w=\\var{per2}$ (Equation 2) (by considering the perimeter of a rectangle)
\n\nReplacing $l$ in Equation 2 by $\\var{ratio2}w$ from Equation 1, we obtain:
\n$2(\\var{ratio2}w)+2w=\\var{per2}$
\n$\\var{2*ratio2+2}w=\\var{per2}$
\n$w=\\var{precround(per2/(2*ratio2+2),3)}$ m
\n\nSubstituting into Equation 1 gives:
\n$l=\\var{ratio2}\\times \\var{precround(per2/(2*ratio2+2),3)} =\\var{precround(ratio2*per2/(2*ratio2+2),3)}$ m
\n\nAnd area = $l \\times w$ = $\\var{precround(ratio2*per2/(2*ratio2+2),3)}\\times\\var{precround(per2/(2*ratio2+2),3)}=\\var{precround((ratio2*per2/(2*ratio2+2))*(per2/(2*ratio2+2)),3)}$ m$^2$
\n\n(b)
\nThe length of a rectangle is $\\var{lent3}$% greater than its width. If the perimeter of the rectangle is $\\var{per3} m$, find the length and width.
\n\nLet the width be $w$ and the length be $l$. Then from the information in the question we obtain the following:
\n$l=\\var{1+lent3/100}w$ (Equation 1)
\n$2l+2w=\\var{per3}$ (Equation 2) (by considering the perimeter of a rectangle)
\n\nReplacing $l$ in Equation 2 by $\\var{1+lent3/100}w$ from Equation 1, we obtain:
\n$2(\\var{1+lent3/100}w)+2w=\\var{per3}$
\n$\\var{2*(1+lent3/100)+2}w=\\var{per3}$
\n$w=\\var{precround(per3/(2*(1+lent3/100)+2),3)}$ m
\n\nSubstituting into Equation 1 gives:
\n$l=\\var{(1+lent3/100)}\\times \\var{precround(per3/(2*(1+lent3/100)+2),3)} =\\var{precround((1+lent3/100)*per3/(2*(1+lent3/100)+2),3)}$ m
", "variables": {"ans23": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ans23", "definition": "ans21*ans22"}, "area1": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "area1", "definition": "random(402..495#0.5)"}, "ans43": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ans43", "definition": "ans41*ans42"}, "lent1": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "lent1", "definition": "random(15..35)"}, "red1": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "red1", "definition": "random(2..10)"}, "ans41": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ans41", "definition": "(per4/2)/(1+ratio4)"}, "ans12": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ans12", "definition": "ans11*lent1"}, "per3": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "per3", "definition": "random(1.2..3.6#0.2)"}, "ans31": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ans31", "definition": "(per3/2)/(2+(lent3/100))"}, "ans21": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ans21", "definition": "(per2/2)/(1+ratio2)"}, "ratio2": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ratio2", "definition": "random(1.5..4.5#0.5)"}, "per4": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "per4", "definition": "random(12.2..24.8#0.2 except[13..24])"}, "ratio4": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ratio4", "definition": "random(1.25..4.75#0.5)"}, "lent3": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "lent3", "definition": "random(15..35#5)"}, "ans11": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ans11", "definition": "(area1+(red1*lent1))/lent1"}, "ans32": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ans32", "definition": "ans31*(1+(lent3/100))"}, "per2": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "per2", "definition": "random(32..80#4)"}, "ans42": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ans42", "definition": "ratio4*ans41"}, "ans22": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ans22", "definition": "ratio2*ans21"}}, "rulesets": {}, "parts": [{"extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "steps": [{"extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "showFeedbackIcon": true, "unitTests": [], "marks": 0, "type": "information", "prompt": "Let the width be $w$ and the length be $l$. Now use the information in the question to set up 2 simultaneous equations involving $w$ and $l$.
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\nLength: [[1]]$m$
\nArea: [[2]]$m^2$
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