1.2 Simultaneous Equations Worksheet

Quick Notes & Formulae

Key concepts for solving systems of equations.

Substitution

Make one variable the subject (e.g.,

y = \dots

) and substitute into the other equation.

Linear & Quadratic

Always rearrange the linear equation first. Substituting quadratic into linear is harder.

Discriminant

Δ = b^{2} - 4 a c

determines intersections of

y = m x + c

and curve.

Tangents

If a line is tangent to a curve, their combined quadratic has

Δ = 0

Intersection Points

Solve for

x

, then find corresponding

y

. Coordinates are

(x_{1}, y_{1})

(x_{2}, y_{2})

Elimination

Useful for two linear equations. Add/Subtract equations to remove a variable.

Past Paper Style Questions

Question 1: Linear and Quadratic

Solve the simultaneous equations:

y = 2 x - 1

y = x^{2} - 3 x + 5

Question 2: Rearranging Required

Solve:

3 x - y = 7

x^{2} + x y + 2 = 0

Question 3: Line and Circle

Find the points of intersection of the line

y = x + 2

and the circle

x^{2} + y^{2} = 10

Question 4: Tangent Condition

Find the value of

k

for which

y = 3 x + k

is a tangent to the curve

y = 2 x^{2} - 5 x + 4

Question 5: Proving No Intersection

Show that the line

y = 2 x - 5

does not intersect the curve

y = x^{2} + 2 x + 1

Question 6: Word Problem (Rectangles)

A rectangle has a perimeter of 20 cm and an area of 24 cm

^{2}

. Find the length and width defining

l > w

Question 7: Word Problem (Triangle)

In a right-angled triangle, the sum of the perpendicular sides is 17 cm. The hypotenuse is 13 cm. Find the area of the triangle.

Question 8: Physics Context

A particle A moves on line

y = t

. Particle B moves on curve

y = t^{2} - 4 t + 6

. At what times

t

are they at the same 'height' (y-value)?

Question 9: Advanced Substitution

Solve the simultaneous equations:

y = x + 2

y^{2} = 4 x + 12

Question 10: Chord Length

The line

y = x - 2

cuts the circle

x^{2} + y^{2} = 8

at points A and B. calculate the length of the chord AB.

Question 11: Sum of Squares

The sum of the perimeters of two squares is 72 cm and the sum of their areas is 162 cm

^{2}

. Find the side length of each square.

Question 12: Circles Radii

The sum of the circumferences of two circles is

50 π

cm and the sum of their areas is

325 π

^{2}

. Find the radii of the two circles.

Question 13: Cuboid Dimensions

A cuboid has dimensions

x

cm,

y

cm and

(y + 1)

cm. Given that

x + y = 17

and the volume of the cuboid is 360 cm

^{3}

, find the values of

x

and

y

Question 14: Composite Solid

A solid is formed by joining a hemisphere of radius

r

cm to a cylinder of the same radius and height

h

cm. The total height of the solid is 20 cm and the curved surface area is

280 π

^{2}

. Find the values of

r

and

h

Question 15: Line Cutting Curve

The line

y = 4 - x

cuts the curve

y = x^{2} - 2 x

at the points

A

and

B

.
(a) Find the coordinates of

A

and

B

.
(b) Find the length of

A B

Question 16: Intersecting Line and Circle

The line

3 x + y = 7

intersects the curve

x^{2} + y^{2} - 6 x + 2 y = 0

at the points

A

and

B

. Find the midpoint of

A B

Question 17: Perpendicular Bisector

The line

2 x - y = 1

cuts the curve

x^{2} + y^{2} = 10

at the points

A

and

B

. Find the equation of the perpendicular bisector of

A B

Question 18: Above the x-axis

The straight line

y = 2 x + 1

intersects the curve

x^{2} - y = 3

at the points

A

and

B

. Given that

A

lies above the x-axis, find the coordinates of

A

and

B

Question 19: Length of Chord (Linear)

The line

x - y + 2 = 0

intersects the curve

x^{2} + y^{2} - 4 x - 6 y + 4 = 0

at the points

A

and

B

. Find the length of

A B

Question 20: Number Split

Split the number 24 into two parts such that the difference between the squares of the two parts is 192. Find the two parts.