IGCSE · Practice questions · Simultaneous equations

Match, eliminate, substitute.

Six original Cambridge-style questions on simultaneous equations, linear and (Extended) linear with non-linear.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Match, then add or subtract.

Match: multiply so one unknown has the same coefficient in both.
Signs: different signs add; same signs subtract.
Substitution: best when one equation already says $y = \dots$.
Back-substitute: find the second unknown, then check both equations.
Non-linear (Extended): substitute the linear into the other and solve the quadratic; give pairs.

Recall

[2 marks]

Solve $x + y = 10$ and $x - y = 4$.

Add: $2x = 14$, $x = 7$ ✓
$y = 3$ ✓

Calculation

[3 marks]

Solve $4x + 3y = 25$ and $2x + 5y = 23$.

Double the second: $4x + 10y = 46$ ✓
Subtract the first: $7y = 21$, $y = 3$ ✓
$2x + 15 = 23$, $x = 4$ ✓

Calculation

[3 marks]

Solve $y = 3x - 4$ and $2x + 3y = 21$.

$2x + 3(3x - 4) = 21$ ✓
$11x - 12 = 21$, $x = 3$ ✓
$y = 9 - 4 = 5$ ✓

Analysis

[4 marks]

5 pens and 2 rulers cost $3.70. 3 pens and 4 rulers cost $3.90.

Find the cost of one pen and the cost of one ruler.

$5p + 2r = 3.70$ and $3p + 4r = 3.90$ ✓
Double the first: $10p + 4r = 7.40$; subtract the second: $7p = 3.50$ ✓
$p = 0.50$ ✓
$2r = 3.70 - 2.50 = 1.20$, $r = 0.60$. Pen $0.50, ruler $0.60. ✓

Calculation

[4 marks]

(Extended) Solve $y = 2x - 1$ and $y = x^{2} - 4$.

$x^{2} - 4 = 2x - 1$ ✓
$x^{2} - 2x - 3 = 0$, $(x - 3)(x + 1) = 0$ ✓
$x = 3,\ y = 5$ ✓
$x = -1,\ y = -3$ ✓

Analysis

[6 marks]

(Extended) The line $y = x + 1$ meets the circle $x^{2} + y^{2} = 25$ at the points A and B.

(a) Find the coordinates of A and B.

(b) Find the length of AB.

(a) $x^{2} + (x + 1)^{2} = 25$, so $2x^{2} + 2x - 24 = 0$ ✓
$x^{2} + x - 12 = 0$, $(x + 4)(x - 3) = 0$ ✓
A$(3, 4)$ and B$(-4, -3)$ ✓
(b) $AB = \sqrt{(3 - (-4))^{2} + (4 - (-3))^{2}} = \sqrt{49 + 49}$ ✓
$= \sqrt{98} = 9.90$ (3 s.f.) ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.