Equations

Expand the brackets and simplify $(x-1)(x^2 + x + 1)$.

May/June 2015

Solve the pair of simultaneous equations $3x = 4y$ and $1 + 5x = 6y$.

May/June 2016

A bottle filled with liquid has a combined mass of $1.27\text{ kg}$. If the bottle is only half-full of liquid, the combined mass is $900$ grams. Calculate the mass of the bottle.

May/June 2016

Solve the pair of simultaneous equations $6x + y = 1$ and $4x - y = 4$.

May/June 2016

Fully factorise $8x^2y-12x^5$.

May/June 2016

Solve the simultaneous equations below: $2x + 3y = 5$ $3x - y = -9$.

May/June 2017

Solve the equation $\frac{6}{x+1} = \frac{5}{x-3}$.

May/June 2017

Carl spent $t$ minutes doing his English homework. His Mathematics homework took three times as long as his English homework. Altogether, he spent $2$ hours $20$ minutes on his English and Mathematics homework.

May/June 2017

Solve the pair of simultaneous equations $5x - 2y = 16$ and $3x + 4y = 7$.

May/June 2017

Express as a single fraction, in its simplest form, $\frac{1}{2x}+\frac{2}{5x}$.

May/June 2017

Demonstrate that $\frac{12}{x + 2} + \frac{10}{x - 1} = \frac{7}{2}$ can be reduced to the equation $7x^2 - 37x - 30 = 0$.

May/June 2018

$A=\begin{pmatrix}4&-1\\2&0\end{pmatrix}$ and $B=\begin{pmatrix}6&-3\\0&-2\end{pmatrix}$.

May/June 2018

Solve the equation $4(p-3) = 2p + 7$.

May/June 2018

The diagram represents the net of an open box with height $3\text{ cm}$. The base of the box has area $15\text{ cm}^2$. The rectangular base has length $x\text{ cm}$. The net’s total area is $A\text{ cm}^2$.

May/June 2018

Solve the equation $3(x + 10) = 12 - 7x$.

May/June 2018

Solve these simultaneous equations: $2x + 3y = 4$ and $3x + 2y = 11$.

May/June 2019

Solve the equation $(2x + 1)(x + 4) = 22$.

May/June 2019

Solve the simultaneous equations, and show the working you use. $9x+4y=-5$ $6x-2y=6$

May/June 2019

For the simultaneous equations below, Solve and Show your working. $x + 6y = 0$ and $3x - 2y = 10$.

May/June 2021

Solve these simultaneous equations. Show your working.

May/June 2021

Express $x^2 + 10x + 6$ in the form $(x + a)^2 + b$.

May/June 2021

If apples are priced at $x$ per kilogram and oranges at $y$ per kilogram, the cost of $5\text{ kg}$ of apples plus $10\text{ kg}$ of oranges is $\$40$. Show that $x + 2y = 8$.

May/June 2022

Solve the simultaneous equations: $x + 2y = 7$ and $3x + 4y = 11$. Show your working.

May/June 2023

The Bukhari family and the Garcia family are heading off on holiday. In the Bukhari family there are 2 adults and 3 children, whereas in the Garcia family there are 4 adults and 1 child.

May/June 2023

Solve this.

May/June 2024

Solve the simultaneous equations, showing all your working.

May/June 2024

A box contains red pens, blue pens and black pens. There are $x$ red pens. The number of blue pens is $5$ greater than the number of red pens. The number of black pens is $2$ times the number of blue pens.

May/June 2024

Company $A$ charges $\$0.50$ per box, together with a fixed charge of $\$125$. Company $B$ charges only a fixed amount of $\$350$. Find how many boxes are moved when Company $A$ and Company $B$ charge the same.

May/June 2024

For a small box, the mass is $x\text{ kg}$. For a large box, the mass is $y\text{ kg}$.

May/June 2025

Solve for $x$ in $5(4-x)=35$.

May/June 2025

Solve the equation $4x - 7 = 9$.

May/June 2025

The quadrilaterals $ABCD$ and $EFGD$ are rectangles.

May/June 2025

The diagram represents a cuboid. Its length is $x$ cm. Its height is $3$ times its length. Its width is $4$ cm less than its length.

May/June 2025

Solve the equation $(x + \frac{7}{2})^2 = \frac{\sqrt{5}}{2}$. Provide both solutions correct to $2$ decimal places.

Oct/Nov 2015

Solve the pair of simultaneous equations $2x + 5y = 2$ and $3x + 4y = -4$.

Oct/Nov 2016

Solve the simultaneous equations $3x+y=9$ together with $2x+3y=-8$.

Oct/Nov 2016

Solve for $x$ in the equation $\dfrac{2x-1}{4} + \dfrac{x-2}{3} = 2$.

Oct/Nov 2016

On Monday, Abdul sold $140$ boxes of matches for $30$ cents each.

Oct/Nov 2016

Solve the pair of simultaneous equations $2y=x+12$ and $3y=11-2x$.

Oct/Nov 2017

A rectangular picture $ABCD$ sits within a rectangular frame. Its length $AB$ is three times the height $x\,\text{cm}$. The frame has width $2\,\text{cm}$.

Oct/Nov 2017

Simplify the expression $7 - 3(5x - 2)$.

Oct/Nov 2019

Find the simultaneous equations $x + 4y = 1$ and $3x + 2y = 8$.

Oct/Nov 2019

Solve for $x$ in $6 + 8x = 7 - 2x$.

Oct/Nov 2019

Find the value of $x$ from the equation $9 - 5x = 2x - 12$.

Oct/Nov 2020

Solve the simultaneous equations $3x - 2y = 12$ and $4x + y = 5$.

Oct/Nov 2020

Solve $6x - 7 > 5 - 2x$.

Oct/Nov 2020

Average speed together with quadratic equations.

Oct/Nov 2020

Find the simultaneous equations and show every step of your working.

Oct/Nov 2021

Show that the equation reduces to $5x^2 + 30x - 39 = 0$.

Oct/Nov 2021

A mass of $4$ cards together with $3$ envelopes is $85\,g$. A mass of $2$ cards together with $5$ envelopes is $67\,g$. Set up a pair of simultaneous equations and solve them to determine the mass of one card and the mass of one envelope.

Oct/Nov 2021

Express $x^2 - 6x - 7$ in the form $(x + a)^2 + b$. Determine the value of $a$ and the value of $b$.

Oct/Nov 2022

Solve $\dfrac{3x-1}{6} + \dfrac{x+2}{4} = \dfrac{5}{3}$.

Oct/Nov 2022

Kate is thinking of a number $n$.

Oct/Nov 2022

A bag has $x$ five-cent coins and $y$ ten-cent coins. Altogether, there are $130$ coins in the bag. Write down an equation, in terms of $x$ and $y$, for the total number of coins in the bag.

Oct/Nov 2022

Solve the simultaneous equations, and show each stage of your working.

Oct/Nov 2023

$\begin{pmatrix} x & 3 \\ 2 & x+1 \end{pmatrix} \begin{pmatrix} x-1 \\ 2 \end{pmatrix} = \begin{pmatrix} 2x+6 \\ y \end{pmatrix}$.

Oct/Nov 2023

Solve the equation $5x + 6 = 3x$.

Oct/Nov 2023

The diagram contains a small rectangle positioned inside a larger rectangle. The larger rectangle has height $x$ cm. Its length is $4$ times its height. The shaded border has width $3$ cm. The area of the small rectangle is $80\,\text{cm}^2$.

Oct/Nov 2023

Solve the simultaneous equations $3a+b=-4$ and $2a+3b=9$. Show all working.

Oct/Nov 2024

Bag $A$ has red balls and green balls. Altogether, the bag contains $x$ balls. The number of green balls is $6$ greater than the number of red balls.

Oct/Nov 2024

Apples cost $x$ cents per kilogram, and Mina pays $\$9$ altogether for apples.

Oct/Nov 2024

A rectangle and a triangle are shown in the diagram. The rectangle’s length is x cm, and its area is 30 cm^{2}. The triangle’s sides measure (x-1) cm, (x+2) cm and (2x-3) cm.

Oct/Nov 2025

Ade and Maha each make cards. On one day, each of them spends 4 hours making cards. Ade needs x minutes for one card.

Oct/Nov 2025

Solve this.

Oct/Nov 2025

Solve the simultaneous equations. You must show all your working. $2a+5b=1.5$ and $3a-10b=42.5$.

Oct/Nov 2025

Rearrange the formula so that a is the subject. $w=\frac{2a+5}{3a-1}$.

Oct/Nov 2025

Solve the equation $\frac{x}{x+3}-\frac{4}{x-5}=1$.

Oct/Nov 2025

Solve the equation $\frac{2x+1}{x+4}+5=x$. Show every step in your working and give your answers correct to 2 decimal places.

Oct/Nov 2025