Simultaneous Equations (Edexcel A Level Maths: Pure): Exam Questions

Solve

$y=2x-1$

$x^2+y^2-2=0$

Solve

$y=x+3$

$2x^2-y^2=5x+3$

Given

$3x^2+4y=83$
$3x+2y=-11$

show that

$x^2-2x-35=0$

Hence solve

$3x^2+4y=83$
$3x+2y=-11$

Given

$x^2+10x+y^2=-20$
$y=2x+10$

show that

$x^2+10x+24=0$

Hence solve

$x^2+10x+y^2=-20$
$y=2x+10$

Given that

$4x-ky=13$
$3x+2ky=7$

where $k$ is a constant, show that

$x=3$

Find an expression for $y$ in terms of $k$.

Given that $y=3$, find $k$.

Use algebra to solve

$x+3y-1=0$

$x^2+9y=2-y^2$

Solve

$x+y=4$
$x^2-4x-3y=0$

Simultaneous equations in $x$ and $y$ are given by

$3kx+y=-4$
$2kx-4y=2$

where $k$ is a non-zero constant.

Solve the equations for $x$ and $y$, giving your answer in terms of $k$ where necessary.

Find the value of $k$ for which the $x$ and $y$ solutions are equal.

Solve

$4x^2+2x-6y=4$
$2x-3y=-1$

Use algebra to solve

$y-x-1=0$

$(2x+1{)}^2-3y^2+3x-10=0$

Simultaneous equations in $x$ and $y$ are given by

$x^2-2y=10$
$2x-y=k$

where $k$ is a constant.

Show that

$x^2-4x+2(k-5)=0$

Find the value of $k$ for which the simultaneous equations have only one $x$ solution and one $y$ solution.

Using the value of $k$ in part (b), solve the simultaneous equations.

Given

$x^2-8y=-40$
$3x+2y=4$

show that

$x^2+12x+24=0$

Hence solve

$x^2-8y=-40$
$3x+2y=4$

giving solutions for $x$ and $y$ in the form

$a\pm b\sqrt{3}$

where $a$ and $b$ are integers to be found.

Given

$4x^2+5xy+9y^2=36$
$y=\frac{2}{3}x+2$

show that 

$x^2+kx=0$

where $k$ is a constant to be found.

Hence solve

$4x^2+5xy+9y^2=36$
$y=\frac{2}{3}x+2$

Given

$15x^2-4y^2=12$
$4x+2y=1$

show that

$x^2-8x+13=0$

Hence solve

$15x^2-4y^2=12$
$4x+2y=1$

giving solutions for $x$ and $y$ in the form

$a\pm b\sqrt{3}$

where $a$ and $b$ are constants to be found.

Simultaneous equations in $x$ and $y$ are given by

$x^2-y+3x=k$
$20x-4y=-5$

where $k$ is a constant.

The simultaneous equations have exactly one $x$ solution and one $y$ solution.

Find the value of $k$.

Using the value of $k$ in part (a), solve the simultaneous equations.

You are asked to advise a client on which parcel delivery service to use to deliver parcels of differing sizes. 

• Linear Deliveries charge a flat rate of £2.25 per parcel, plus 40p times the mass of the parcel in kilograms

• Square Deal Deliveries charge a flat rate of £4 per parcel, plus 1p times the square of the parcel’s mass in kilograms

Form and solve simultaneous equations to find the two weights of parcels that are the same price in both companies.

Hence, find

• the range of weights of parcels for which Linear Deliveries is cheaper

• the range of weights of parcels for which Square Deal Deliveries is cheaper

A firework is launched inside a large shed with a sloping roof. 

The horizontal distance from the point it was launched, $x$ metres, and the height of the firework, $h$ metres, can be modelled by

 $h=0.84x-0.07x^2$

The sloping roof of the shed can be modelled by

 $h=2+0.1x$

According to the model, determine whether or not the firework hits the roof. Show your full working.

Solve

$9x^2-7xy+4y^2=36$
$3x+2y=-6$

Given

$8y^2-3x^2-4x=-\frac{11}{2}$
$3x+4y=1$

 show that

$3x^2-14x+12=0$

Hence solve

$8y^2-3x^2-4x=-\frac{11}{2}$
$3x+4y=1$

giving solutions for $x$ and $y$ in the form

$a\pm b\sqrt{c}$

where $a$ and $b$ are rational numbers to be found and $c$ is a prime number to be found.

Simultaneous equations in $x$ and $y$ are given by

$5x+(k+1)y=-20$
$7x-2ky=2y+6$

where $k$ is a constant.

Solve the equations for $x$ and $y$, giving your answers in terms of $k$ where necessary.

Find the value of $k$ for which there are no solutions to the simultaneous equations.

Simultaneous equations in $x$ and $y$ are given by

$x^2+2y^2=25$
$x-y=k$

where $k$ is a constant.

Find the possible values of $k$ for which

(i) exactly one $x$ solution exists

(ii) two distinct $x$ solutions exist

(iii) no $x$ solutions exist