// Numbas version: finer_feedback_settings {"name": "Math Diagnostic - Sep 26", "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": true, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], []], "questions": [{"name": "Algebra vocabulary", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}], "rulesets": {}, "functions": {}, "ungrouped_variables": ["test"], "metadata": {"description": "

A fill-in-the-blank style question to test vocabulary within the topic of Algebra.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "variable_groups": [], "advice": "

The words and their definitions are as follows:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

Word	Definition	Example
Simplify	Put into its simplest form.	The following equation, $2a +3a = 5a$ has been simplified . Due to the collection of like terms.
Solve	To obtain a solution to a mathematical problem.	Solve $5x = 30$, obtaining the solution $x=6$.
Calculate	You will need to do a sum to reach the answer.	5 students measured their heights; calculate the average of their heights.
Variable	An unknown value that is represented by a letter.	The variable x is unknown.
Equation	Will always have an equals sign and both sides are equal.	Jane buys 2 cups and 3 mugs. The total number of cups and mugs can be expressed using the following equation: $ 2+3=5$.
Formula	A way of showing a known equation using numbers and letters.	The formula for the area of a triangle is $\\frac{\\mathrm{base} \\times \\mathrm{height}}{2}$.
Expand	Removing the brackets.	The following expression has been expanded: $(x+3)(x-1) = x^2 +2x - 3$.
Factorise	To put in brackets, finding a common factor; this is usually the simplest form.	The expression $2x^2 +4x -6 = 2(x-1)(x+3)$ has been factorised.
Linear	An expression or equation that is plotted as a straight line.	An expression or equation that is plotted as a straight line is said to be linear.
Quadratic	An expression or equation that is plotted as a parabola curve. Will include an $x^2$ term.	For example, $y = x^2+2x-3$.
Term	A component of an expression or equation.	$5y$ is a term in $3x+5y$.
Coefficient	The number in front of a variable to make a term.	In $2x$, $2$ is the coefficient of $x$.
Parabola	The 'U shaped' curve on a quadratic graph.	A banana shape can be described as a parabola

", "statement": "

Fill the gaps in the following sentences.

In each case, choose the word which best fits in the sentence.

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Only here to be able to publish the question.

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The following equation, $2a +3a = 5a$ has been [[0]], due to the collection of like terms.

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Simplified

", "

Expanded

", "

Factorised

", "

Solved

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[[0]] $5x = 30$, obtaining the solution $x=6$.

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Simplify

", "

Calculate

", "

Divide

", "

Solve

5 students measured their heights. [[0]] the average of their heights.

", "gaps": [{"variableReplacementStrategy": "originalfirst", "choices": ["

Simplify

", "

Calculate

", "

Divide

", "

Solve

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The [[0]] x is unknown.

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Symbol

", "

Term

", "

Formula

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Variable

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Jane buys 2 cups and 3 mugs. The total number of cups and mugs can be expressed using the following [[0]]: $ 2+3=5$.

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Variable

", "

Equation

", "

Formula

", "

Linear

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The [[0]] for the area of a triangle is $\\frac{\\mathrm{base} \\times \\mathrm{height}}{2}$.

", "gaps": [{"variableReplacementStrategy": "originalfirst", "choices": ["

Solution

", "

Equation

", "

Formula

", "

Definition

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The following expression has been [[0]] $(x+3)(x-1) = x^2 +2x - 3$.

", "gaps": [{"variableReplacementStrategy": "originalfirst", "choices": ["

Multiplied

", "

Expanded

", "

Factorised

", "

Solved

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The expression $2x^2 +4x -6 = 2(x-1)(x+3)$ has been [[0]]

", "gaps": [{"variableReplacementStrategy": "originalfirst", "choices": ["

Simplified

", "

Expanded

", "

Factorised

", "

Calculated

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An expression or equation that is plotted as a straight line is said to be [[0]]

", "gaps": [{"variableReplacementStrategy": "originalfirst", "choices": ["

Linear

", "

Expanded

", "

Quadratic

", "

Parabola

A [[0]] equation will include an $x^2$ term. For example, $y = x^2+2x-3$.

", "gaps": [{"variableReplacementStrategy": "originalfirst", "choices": ["

Linear

", "

Quadratic

", "

Formula

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Function

$5y$ is a [[0]] in $3x+5y$.

", "gaps": [{"variableReplacementStrategy": "originalfirst", "choices": ["

Variable

", "

Term

", "

Coefficient

", "

Symbol

In $2x$, $2$ is the [[0]] of $x$.

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Constant

", "

Coefficient

", "

Variable

", "

Term

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The 'U shaped' curve on a quadratic graph is called a [[0]].

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Quadratic

", "

Linear

", "

Hyperbola

", "

Parabola

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Solve a simple linear equation algebraically. The unknown appears on both sides of the equation.

", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["d", "f", "g", "h", "x", "gcd_hfdg", "hf_coprime", "dg_coprime", "finalb"], "type": "question", "rulesets": {}, "variable_groups": [], "statement": "", "advice": "

We are asked to solve the equation

\\[ \\var{d}x-\\var{f}=\\var{g}x+\\var{h} \\]

In this equation, there are $x$ terms and constant terms on both sides of the equals sign.

To solve this equation, we must rearrange it to get $x$ on its own.

\\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}

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$\\var{d}x-\\var{f}=\\var{g}x+\\var{h}$

What is the value of $x$?

$x = $ [[0]]

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Solve a linear equation of the form $ax+b = c$, where $a$, $b$ and $c$ are integers.

The answer is always an integer.

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We need to solve the equation

\\[ \\var{a}x+\\var{b}=\\var{c} \\]

To solve this equation, we must rearrange the equation to put $x$ on its own.

To do this, we should subtract $\\var{b}$ from both sides and then divide through by $\\var{a}$ to get the value for $x$.

\\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}

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$\\var{a}x+\\var{b}=\\var{c}$

What is the value of $x$?

$x = $ [[0]]

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This question is made up of 10 exercises to practice the multiplication of brackets by a single term.

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Expand the expressions below by multiplying each of the terms inside the brackets by the term outside. Give each answer in its simplest form.

", "advice": "

Expand brackets using the general formula $\\displaystyle a(x+c)=ax+ac$. This means we multiply each term inside the brackets by the term outside the brackets.

It is easy to forget that the sign outside the brackets also needs to be involved in the multiplication so remember that when two of the same sign are multiplied, the resultant term is positive and when opposite signs are multiplied, the result is negative.

a)

\\[
\\begin{align}
\\simplify[terms]{{a[1]}({a[2]}x+{a[3]})}&=
\\simplify[!collectNumbers]{({a[1]}{a[2]})x+({a[1]}{a[3]})}\\\\&
=\\simplify{{a[1]}*{a[2]}x+{a[1]}{a[3]}}\\text{.}
\\end{align}
\\]

b)

\\[
\\begin{align}
\\simplify[terms]{{a[4]}({a[5]}x+{a[6]})}&=
\\simplify[!collectNumbers]{{a[4]}{a[5]}x+{a[4]}{a[6]}}\\\\&=
\\simplify{{a[4]}*{a[5]}x+{a[4]}{a[6]}}\\text{.}
\\end{align}
\\]

c)

\\[
\\begin{align}
\\simplify[terms]{{a[7]}({a[8]}x^2+{a[9]}y)}&=
\\simplify[!collectNumbers]{{a[7]}{a[8]}x^2+{a[7]}{a[9]}y}\\\\&=
\\simplify{{a[7]}*{a[8]}x^2+{a[7]}*{a[9]}y}\\text{.}
\\end{align}
\\]

d)

\\[
\\begin{align}
\\simplify[terms]{{a[10]}({a[11]}x^2+{a[12]}y)}&=
\\simplify[!collectNumbers]{{a[10]}{a[11]}x^2+{a[10]}{a[12]}y}\\\\&=
\\simplify{{a[10]}*{a[11]}x^2+{a[10]}*{a[12]}y}\\text{.}
\\end{align}
\\]

e)

\\[
\\begin{align}
\\simplify[terms]{{a[13]}x({a[14]}x^2+{a[15]}x+{a[16]})}&=
\\simplify[!collectNumbers]{{a[13]}x{a[14]}x^2+{a[13]}x{a[15]}x+{a[13]}x{a[16]}}\\\\&=
\\simplify{{a[13]}{a[14]}x^3+{a[13]}{a[15]}x^2+{a[13]}{a[16]}x}\\text{.}
\\end{align}
\\]

f)

\\[
\\begin{align}
\\simplify[terms]{{a[17]}x({a[18]}x^2+{a[19]}x+{a[20]})}&=
\\simplify[!collectNumbers]{{a[17]}x{a[18]}x^2+{a[17]}x{a[19]}x+{a[17]}x{a[20]}}\\\\&=
\\simplify{{a[17]}{a[18]}x^3+{a[17]}{a[19]}x^2+{a[17]}{a[20]}x}\\text{.}
\\end{align}
\\]

g)

\\[
\\begin{align}
\\simplify[terms]{{a[21]}x({a[22]}x^2+{a[23]}x)+{a[24]}x^2+{a[25]}x^3}&=
\\simplify[!collectNumbers]{x^2({a[21]}{a[23]})+x^2{a[24]}+x^3({a[21]}{a[22]})+x^3{a[25]}}\\\\&=
\\simplify[!collectNumbers]{x^2({a[21]}{a[23]}+{a[24]})+x^3({a[21]}{a[22]}+{a[25]})}\\\\&=
\\simplify{x^2({a[21]}{a[23]}+{a[24]})+x^3({a[21]}{a[22]}+{a[25]})}\\text{.}
\\end{align}
\\]

h)

\\[
\\begin{align}
\\simplify[terms]{({a[26]}x^2+{a[27]}x^3)+{a[28]}x({a[29]}x^2+{a[30]}x)}&=
\\simplify[!collectNumbers]{x^2({a[26]})+x^2({a[28]}{a[30]})+x^3({a[28]}{a[29]})+x^3({a[27]})}\\\\&=
\\simplify[!collectNumbers]{x^2({a[26]}+{a[28]}{a[30]})+x^3({a[28]}{a[29]}+{a[27]})}\\\\&=
\\simplify{x^2({a[26]}+{a[28]}{a[30]})+x^3({a[28]}{a[29]}+{a[27]})}\\text{.}
\\end{align}
\\]

i)

\\[
\\begin{align}
\\simplify[terms]{{a[31]}({a[32]}x+{a[33]}y)+{a[34]}x({a[42]}+{a[35]}y)}&=
\\simplify[!collectNumbers]{({a[31]}{a[32]})x+({a[34]}{a[42]})x+{a[31]}{a[33]}y+{a[34]}{a[35]}x*y}\\\\&=
\\simplify[!collectNumbers]{({a[31]}{a[32]}+{a[34]}{a[42]})x+{a[31]}{a[33]}y+{a[34]}{a[35]}x*y}\\\\&=
\\simplify{({a[31]}{a[32]}+{a[34]}{a[42]})x+{a[31]}{a[33]}y+{a[34]}{a[35]}x*y}\\text{.}
\\end{align}
\\]

j)

\\[
\\begin{align}
\\simplify[terms]{{a[36]}a^2({a[37]}+{a[38]}b)+{a[39]}b^2({a[40]}a+{a[41]}b)}&=
\\simplify[!collectNumbers]{{a[37]}{a[36]}a^2+{a[38]}{a[36]}a^2b+{a[40]}{a[39]}a*b^2+{a[39]}{a[41]}b^3}\\\\&=
\\simplify{{a[37]}{a[36]}a^2+{a[38]}{a[36]}a^2b+{a[40]}{a[39]}a*b^2+{a[39]}{a[41]}b^3}\\text{.}
\\end{align}
\\]

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$\\simplify{{a[1]}({a[2]}x+{a[3]})}=$ [[0]]

It doesn't look like you've expanded - make sure you don't use any brackets in your answer.

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$\\simplify{{a[4]}({a[5]}x+{a[6]})}=$ [[0]]

It doesn't look like you've expanded - make sure you don't use any brackets in your answer.

$\\simplify{{a[7]}({a[8]}x^2+{a[9]}y)}=$ [[0]]

It doesn't look like you've expanded - make sure you don't use any brackets in your answer.

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$\\simplify{{a[10]}({a[11]}x^2+{a[12]}y)}=$ [[0]]

It doesn't look like you've expanded - make sure you don't use any brackets in your answer.

$\\simplify{{a[13]}x({a[14]}x^2+{a[15]}x+{a[16]})}=$ [[0]]

It doesn't look like you've expanded - make sure you don't use any brackets in your answer.

$\\simplify{{a[17]}x({a[18]}x^2+{a[19]}x+{a[20]})}=$ [[0]]

It doesn't look like you've expanded - make sure you don't use any brackets in your answer.

$\\simplify{{a[21]}x({a[22]}x^2+{a[23]}x)+{a[24]}x^2+{a[25]}x^3}=$ [[0]]

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$\\simplify{({a[26]}x^2+{a[27]}x^3)+{a[28]}x({a[29]}x^2+{a[30]}x)}=$ [[0]]

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$\\simplify{{a[31]}({a[32]}x+{a[33]}y)+{a[34]}x({a[42]}+{a[35]}y)}=$ [[0]]

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$\\simplify{{a[36]}a^2({a[37]}+{a[38]}b)+{a[39]}b^2({a[40]}a+{a[41]}b)}=$ [[0]]

It doesn't look like you've expanded - make sure you don't use any brackets in your answer.

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Given a description in words of the costs of some items in terms of an unknown cost, write down an expression for the total cost of a selection of items. Then simplify the expression, and finally evaluate it at a given point.

The word problem is about the costs of sweets in a sweet shop.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

{pname} eats a lot of sweets. You are trying to work out the cost of the sweets that {pname} ate last week.

{pname} ate $\\var{a1}$ packets of lollipops, $\\var{b1}$ packets of toffee and $\\simplify{{c1}}$ packets of jelly sweets.

You know that a packet of toffee costs $£1$ more than a packet of lollipops, and a packet of jelly sweets costs half as much as a packet of toffees.

", "advice": "

a)

We are told that the price of a packet of lollipops is represented by the letter $x$.

A packet of toffee costs $£1$ more than a packet of lollipops, i.e. $x+1$.

A packet of jelly sweets costs half as much as a packet of toffee, so $\\frac{1}{2}(x+1)$.

b)

To find the total cost, multiply the expressions above for the cost of each kind of sweet by the number of packets eaten, and add them together.

Without simplifying, we obtain:

\\begin{align}
\\text{Cost} &= \\simplify[]{{a1}x+{b1}(x+1) + {c1}*(1/2)*(x+1)} \\\\
&= \\simplify[]{{a1}x+{b1}(x+1) + {c1/2}*(x+1)}
\\text{.}
\\end{align}

c)

The first step in simplifying this expression is to expand both sets of brackets:

\\begin{align}
\\simplify[]{ {a1}x + {b1}(x+1) + {c1/2}*(x+1)} &= \\simplify[]{ {a1}x + {b1}x + {b1}*1 + {c1/2}x + {c1/2}*1} \\\\
&= \\simplify[] { {a1}x + {b1}x + {b1} + {c1/2}x + {c1/2} } \\text{.}
\\end{align}

Finally, collect like terms:

\\begin{align}
\\simplify[] { {a1}x + {b1}x + {b1} + {c1/2}x + {c1/2} } &= \\simplify[]{ {a1+b1+c1/2}x + {b1+c1/2} } \\text{.}
\\end{align}

d)

Once we know that the price of a packet of lollipops is $£2$, we can substitute this for $x$ in the equation above.

\\begin{align}
\\text{Cost}&=\\simplify{ {a1+b1+c1/2}x+{b1+c1/2} }\\\\
&=\\var{a1+b1+c1/2} \\times 2+\\var{b1+c1/2} \\\\
&=\\var{(a1+b1+c1/2)*2+b1+c1/2} \\text{.}
\\end{align}

So {pname} spent $£\\var{total}$ on sweets last week.

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Number of packets of toffee eaten

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Number of packets of jelly sweets eaten.

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Number of packets of lollipops eaten

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The total spent.

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Let the cost of a packet of lollipops be $£x$.

Write an expression in terms of $x$ for the cost of each kind of sweet:

Lollipops: £[[0]]

Toffees: £[[1]]

Jelly sweets: £[[2]]

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Write an algebraic expression for the overall cost of the sweets {pname} ate, in terms of $x$.

£[[0]]

Now simplify your expression for the total cost.

£[[0]]

You find out that a packet of lollipops costs $£2$.

Calculate {pname}'s total expenditure on sweets last week.

£[[0]]

Don't use brackets

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random variables for part 1

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You must collect like terms to fully simplify.

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$\\var{c[0]}x+\\var{c[1]}x+\\var{c[2]}x=$ [[0]]

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$\\var{a[1]}x^2+\\var{a[2]}x^2+\\var{a[3]}x+\\var{a[4]}x +\\var{a[0]}=$ [[0]]

You must condense your answer to fully simplify.

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$\\var{b[0]}y^5+\\var{b[1]}y^5+\\var{b[2]}y^5+\\var{b[4]}y^5+\\var{b[3]}y^5=$ [[0]]

You must condense your answer to fully simplify.

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$\\var{d[0]}ab+\\var{d[1]}abc+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c+\\var{d[5]}abc=$ [[0]]

You must condense your answer to fully simplify.

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$\\var{f[0]}a^2b+\\var{f[1]}ab^2+\\var{f[2]}ab+\\var{f[3]}a^2b+\\var{f[4]}ab^2=$ [[0]]

You must condense your answer to fully simplify. *'s are not needed to indicate multiplication here.

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$\\var{g[0]}(\\var{g[1]}x+\\var{g[2]}y)+\\var{g[4]}x+\\var{g[5]}y=$ [[0]]

9You should not have brackets in your answer.

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You must condense your answer to fully simplify.

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$\\var{h[0]}x(\\var{h[1]}x+\\var{h[2]}z)+\\var{h[3]}x+\\var{h[6]}z+\\var{h[4]}x^2+\\var{h[5]}z^2=$ [[0]]

You must condense your answer to fully simplify.

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$\\var{j[0]}(\\var{j[1]}x-\\var{j[2]}y)+\\var{j[3]}(\\var{j[4]}x-\\var{j[5]}y)+\\var{j[6]}(\\var{j[7]}x-\\var{j[8]}y)=$ [[0]]

", "unitTests": [], "customName": ""}], "advice": "

When simplifying expressions, only terms of the same type or like terms can be added together.

Algebraic symbols or letters can be added together provided that they are raised to the same power. For example, we can add $x^2+x^2=2x^2$, but we cannot collect both $x^2$ and $x$ into one term.

a)

\\[
\\begin{align}
\\var{c[0]}x+\\var{c[1]}x+\\var{c[2]}x&=(\\var{c[0]}+\\var{c[1]}+\\var{c[2]})x\\\\
&=\\simplify{({c[0]}+{c[1]}+{c[2]})}x
\\end{align}
\\]

b)

\\[
\\begin{align}
\\var{a[1]}x^2+\\var{a[2]}x^2+\\var{a[3]}x+\\var{a[4]}x +\\var{a[0]}&=(\\var{a[1]}+\\var{a[2]})x^2+(\\var{a[3]}+\\var{a[4]})x +\\var{a[0]}\\\\
&=\\simplify{({a[1]}+{a[2]})}x^2+\\simplify{({a[3]}+{a[4]})}x+\\var{a[0]}
\\end{align}
\\]

c)

\\[
\\begin{align}
\\var{b[0]}y^5+\\var{b[1]}y^5+\\var{b[2]}y^5+\\var{b[4]}y^5+\\var{b[3]}y^5&=(\\var{b[0]}+\\var{b[1]}+\\var{b[2]}+\\var{b[4]}+\\var{b[3]})y^5\\\\
&=\\simplify{({b[1]}+{b[2]}+{b[3]}+{b[4]}+{b[0]})}y^5
\\end{align}
\\]

d)

\\[
\\begin{align}
\\var{d[0]}ab+\\var{d[1]}abc+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c+\\var{d[5]}abc
&=(\\var{d[1]}+\\var{d[5]})abc+\\var{d[0]}ab+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c\\\\
&=\\simplify{{d[1]}+{d[5]}}abc+\\var{d[0]}ab+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c
\\end{align}
\\]

e)

\\[
\\begin{align}
\\var{f[0]}a^2b+\\var{f[1]}ab^2+\\var{f[2]}ab+\\var{f[3]}a^2b+\\var{f[4]}ab^2
&=(\\var{f[0]}+\\var{f[3]})a^2b+(\\var{f[1]}+\\var{f[4]})ab^2+\\var{f[2]}ab\\\\
&=\\simplify{{f[0]}+{f[3]}}a^2b+\\simplify{{f[1]}+{f[4]}}ab^2+\\var{f[2]}ab
\\end{align}
\\]

f)

\\[
\\begin{align}
\\var{g[0]}(\\var{g[1]}x+\\var{g[2]}y)+\\var{g[4]}x+\\var{g[5]}y
&=(\\var{g[0]}\\times \\var{g[1]}+\\var{g[4]})x+(\\var{g[0]} \\times\\var{g[2]}+\\var{g[5]})y\\\\
&=(\\simplify{{g[0]}*{g[1]}}+\\var{g[4]})x+(\\simplify{{g[0]}*{g[2]}}+\\var{g[5]})y\\\\
&=\\simplify{{g[0]}*{g[1]}+{g[4]}}x+\\simplify{{g[0]}*{g[2]}+{g[5]}}y
\\end{align}
\\]

g)

\\[
\\begin{align}
\\var{h[0]}x(\\var{h[1]}x+\\var{h[2]}z)+\\var{h[3]}x+\\var{h[6]}z+\\var{h[4]}x^2+\\var{h[5]}z^2
&=(\\simplify[]{{h[0]}{h[1]}}+\\var{h[4]})x^2+(\\simplify[]{{h[0]}{h[2]}})zx+\\var{h[3]}x+\\var{h[5]}z^2+\\var{h[6]}z\\\\
&=(\\simplify{{h[0]}{h[1]}}+\\var{h[4]})x^2+(\\simplify[]{{h[0]}{h[2]}})zx+\\var{h[3]}x+\\var{h[5]}z^2+\\var{h[6]}z\\\\
&=\\simplify{{h[0]}*{h[1]}+{h[4]}}x^2+\\simplify{{h[0]}*{h[2]}}zx+\\simplify{{h[3]}x+{h[5]}}z^2+\\var{h[6]}z
\\end{align}
\\]

h)

\\[
\\begin{align}
\\var{j[0]}(\\var{j[1]}x-\\var{j[2]}y)+\\var{j[3]}(\\var{j[4]}x-\\var{j[5]}y)+\\var{j[6]}(\\var{j[7]}x-\\var{j[8]}y)
&= (\\simplify[]{{j[0]}{j[1]}}+\\simplify[]{{j[3]}{j[4]}}+\\simplify[]{{j[6]}{j[7]}})x-(\\simplify[]{{j[0]}{j[2]}}+\\simplify[]{{j[3]}{j[5]}}+\\simplify[]{{j[6]}{j[8]}})y\\\\
&= (\\simplify{{j[0]}{j[1]}}+\\simplify{{j[3]}{j[4]}}+\\simplify{{j[6]}{j[7]}})x-(\\simplify{{j[0]}{j[2]}}+\\simplify{{j[3]}{j[5]}}+\\simplify{{j[6]}{j[8]}})y\\\\
&= \\simplify{({j[0]}*{j[1]}+{j[4]*j[3]}+{j[6]}*{j[7]})x}-\\simplify{({j[0]}*{j[2]}+{j[5]}{j[3]}+{j[6]}*{j[8]})y}
\\end{align}
\\]

", "tags": ["collecting terms", "expanding brackets", "simplifying algebraic expressions", "simplifying expressions", "taxonomy"], "preamble": {"js": "", "css": ""}, "rulesets": {}, "functions": {}, "ungrouped_variables": [], "statement": "

For each expression below, collect like terms and expand brackets.

The * symbol is required between algebraic symbols, e.g. $5ab^2$ should be written 5*a*b^2.

", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Eight expressions, of increasing complexity. The student must simplify them by expanding brackets and collecting like terms.

"}, "variablesTest": {"condition": "", "maxRuns": 100}}, {"name": "Substitute values into formulas", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}], "metadata": {"description": "

Substitute given values into formulas.

", "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.

a)

As $x = \\var{n+2}$,

substitute $\\var{n+2}$ into $\\var{x2}x^2 + \\var{x1}x + \\var{const}$.

\\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}

As $y = \\var{n}$,

substitute $\\var{n}$ into $\\var{n+1}y^2-\\var{x2}y$.

\\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}

As we are given a temperature in degrees Celcius, $T_C = \\var{T_C}°C.$

Substituting $T_C$ into $T_C = 1.8\\,T_C + 32$.

\\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}

", "variable_groups": [{"name": "Name variables", "variables": ["name", "name2", "pronoun"]}, {"name": "Temperature conversion", "variables": ["T_F", "T_C"]}], "rulesets": {}, "statement": "

Substitute the given values in the equations below.

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A curve is defined by a function $y=\\simplify{{x2}x^2 + {x1}x + {const}}$.

What is the $y$ coordinate value of the point on the curve at $x=\\var{n+2}$?

$y =$ [[0]]

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{name[n]} sells luxury yachts.

The predicted sales of the luxury yachts are defined by

\\[S=\\simplify{{n+1}y^2-{x2}y},\\]

where
$S$ 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.

Calculate $S$, the number of sales predicted this year.

$S =$ [[0]]

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You can convert temperatures from degrees celsius to degrees fahrenheit by using the formula

\\[T_F=1.8\\, T_C+32,\\]

where
$T_F$ = Temperature in $°F$
$T_C$ = Temperature in $°C$.

Convert $\\var{T_C}°C$ into degrees fahrenheit.

$T_F =$ [[0]] $°F$

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Defines the pronoun in the question.

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Creates a random integer value for the temperature in degrees celcius.

", "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

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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.

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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

", "definition": "[\"Andrew\",\"Susan\",\"Tom\",\"Geraldine\",\"Joshua\",\"Chantel\"]", "group": "Name variables", "name": "name2", "templateType": "anything"}, "x1": {"description": "

The x coefficient

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