Simultaneous EquationsEdexcel IGCSE Maths: Revision notes

Section 1

What are simultaneous equations?

Simultaneous equations are two (or more) equations that share the same unknowns, and which must both be true at the same time. Solving them means finding the value(s) of the unknowns that satisfy every equation together.

Graphically, the solution to a pair of equations is the point (or points) where their graphs intersect.

Two straight lines intersect at exactly one point (unless parallel).
A straight line and a curve (e.g. a parabola) can intersect at 0, 1, or 2 points.

Key termssimultaneous equationsintersection

Think of it like this

Think of each equation as a rule. A pair $(x, y)$ only 'wins' if it obeys both rules simultaneously — not just one.

Section 2

How do I solve linear-linear equations by elimination?

Elimination removes one variable by adding or subtracting the equations.

Method:

Multiply one or both equations so that the coefficients of one variable match (same size, ignore sign for now).
Add the equations if the matching coefficients have opposite signs; subtract if they have the same sign.
Solve the resulting single-variable equation.
Substitute back into either original equation to find the second variable.
Check both values in the other original equation.

Example: $3x + 2y = 16$ $5x - 2y = 8$

The $y$ -coefficients are already equal in size with opposite signs, so add: $8x = 24 \Rightarrow x = 3$

Substitute into the first equation: $3(3) + 2y = 16 \Rightarrow 2y = 7 \Rightarrow y = 3.5$

Key termseliminationcoefficient

Exam tip

Same signs → SUBTRACT. Different signs → ADD. Remember 'SSS, DDA' (Same Signs Subtract, Different signs Add).

Common mistake

Forgetting to multiply the WHOLE equation (both sides, every term) when scaling up — a common cause of wrong answers.

Section 3

How do I solve linear-linear equations by substitution?

Substitution works well when one equation is already (or easily) written as $y = ...$ or $x = ...$ .

Method:

Rearrange one equation to make $x$ or $y$ the subject.
Substitute this expression into the other equation.
Solve the resulting equation in one variable.
Substitute back to find the second variable.
Check both values in both original equations.

Example: $y = 2x - 1$ $3x + y = 14$

Substitute: $3x + (2x - 1) = 14 \Rightarrow 5x - 1 = 14 \Rightarrow 5x = 15 \Rightarrow x = 3$

Then $y = 2(3) - 1 = 5$ .

Key termssubstitutionsubject of a formula

Exam tip

Always put brackets around the substituted expression, then expand carefully — this avoids sign errors.

Example

If neither equation gives $x$ or $y$ alone easily, elimination is usually faster than substitution.

Section 4

How do I solve linear-quadratic simultaneous equations?

When one equation is linear and the other is quadratic (contains an $x^2$ or $y^2$ term, or an $xy$ term), substitution is the only method — you cannot eliminate.

Method:

Rearrange the linear equation to make $y$ (or $x$ ) the subject.
Substitute this expression into the quadratic equation.
Expand and rearrange into the form $ax^2 + bx + c = 0$ .
Solve the quadratic (factorise, quadratic formula, or completing the square) — this usually gives two values.
Substitute each value back into the linear equation to find the matching $y$ value(s).
State solutions as coordinate pairs $(x, y)$ .

Example: $y = x + 1$ $x^2 + y^2 = 25$

Substitute: $x^2 + (x+1)^2 = 25$ $x^2 + x^2 + 2x + 1 = 25$ $2x^2 + 2x - 24 = 0$ $x^2 + x - 12 = 0$ $(x + 4)(x - 3) = 0$ $x = -4$ or $x = 3$

When $x = -4$ : $y = -3$ . When $x = 3$ : $y = 4$ .

Solutions: $(-4, -3)$ and $(3, 4)$ .

Key termsquadratic equationcoordinate pair

Common mistake

Substitute the linear expression into the QUADRATIC equation, never the other way round — substituting a quadratic expression into a linear one creates unnecessary powers.

Common mistake

Forgetting to find the second $y$ value — each $x$ solution needs its own matching $y$ , they are not interchangeable.

Section 5

How do I check my answers and interpret the graph?

Always verify your solution(s) by substituting both values back into both original equations — if either fails, recheck your algebra.

Graphically:

Linear-linear: one intersection point (or none if parallel, or infinite if identical lines).
Linear-quadratic: two intersection points if the line crosses the curve, one if the line is a tangent (touches at one point), or none if the discriminant of the resulting quadratic is negative.

If the quadratic formula gives a negative number under the square root, there are no real solutions — the line does not meet the curve.

Key termsdiscriminanttangent

Exam tip

In an exam, always substitute your final answers back in — it takes 30 seconds and catches almost every arithmetic slip.

Must Know

Elimination: same signs → subtract; different signs → add.
Substitution: rearrange one equation, substitute into the other, then solve.
Linear-quadratic pairs MUST be solved by substitution, never elimination.
Substituting the linear expression into the quadratic gives an equation solvable by factorising, completing the square, or the quadratic formula.
Each $x$ value from a linear-quadratic pair has its own matching $y$ value — always find both.
Always check solutions in the original equations before writing your final answer.

That's the notes covered.

Carry on to the next subtopic.