Question bank › IGCSE Maths › Equations, formulae and identities

Simultaneous equations (linear, and linear with quadratic)

Answer on paper first, then reveal the mark scheme one mark at a time.

1Grade 4–5Calculator3 marks

Solve the simultaneous equations

$3x + 4y = 5$
$5x - 2y = 17$

Show clear algebraic working. [3]

for a correct method to eliminate one variable, e.g. $3x + 4y = 5$ and $10x - 4y = 34$ added, with no more than one arithmetic errorM1
(dep) for substituting their found value into one of the equations, or a correct method to eliminate the other variableM1
$x = 3$, $y = -1$A1
Answer: $x = 3$, $y = -1$
Examiner tip: 'Show clear algebraic working' means trial and improvement scores nothing, even with the right answer.

2Grade 8–9HigherCalculator5 marks

Solve algebraically the simultaneous equations

$x^2 + y^2 = 34$
$y = 2x - 1$ [5]

for substituting to get $x^2 + (2x - 1)^2 = 34$ (or substituting $x = \dfrac{y + 1}{2}$)M1
for the correct 3-term quadratic $5x^2 - 4x - 33 = 0$ (or $5y^2 + 2y - 135 = 0$)A1
for a correct method to solve their 3-term quadratic, e.g. $(5x + 11)(x - 3) = 0$M1
$x = 3$ and $x = -2.2$ (or $y = 5$ and $y = -5.4$)A1
$x = 3,\ y = 5$ and $x = -2.2,\ y = -5.4$, correctly pairedA1
Answer: $x = 3, y = 5$ or $x = -2.2, y = -5.4$
Examiner tip: Expand (2x − 1)² as 4x² − 4x + 1 — writing 4x² + 1 loses the A1 for the quadratic and both final accuracy marks.

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