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Linear Simultaneous Equations

Simultaneous equations are two equations that must both be true at once. Solving them finds the pair $(x, y)$ that satisfies both — the point where the two lines cross. There are two standard methods: elimination and substitution.

30 min Video by Zeeshan Zamurred Equations and Inequalities

Edexcel AS level Maths: 3.1 Linear Simultaneous EquationsWatch the full walkthrough before the notes below.

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What you'll be able to do

Solve a pair of linear simultaneous equations by elimination
Solve a pair by substitution
Interpret the solution as the intersection of two lines
Check a solution in both original equations

Method 1: Elimination

Make the coefficients of one variable match (multiplying an equation if needed), then add or subtract the equations to eliminate that variable.

1The

y

coefficients already match, so subtract:

(3 x + y) - (2 x + y) = 11 - 8

2This gives

x = 3

3Substitute back:

2 (3) + y = 8 \Rightarrow y = 2

Answer

x = 3, y = 2

Tip — Same signs → subtract; different signs → add. That eliminates the matched variable cleanly.

Method 2: Substitution

Rearrange one equation to make a variable the subject, then substitute it into the other. This is especially handy when a variable already has coefficient 1.

1Substitute

y = 2 x - 1

into the second:

3 x + (2 x - 1) = 9

2So

5 x - 1 = 9 \Rightarrow x = 2

3Then

y = 2 (2) - 1 = 3

Answer

x = 2, y = 3

What the solution means

Each linear equation is a straight line. The simultaneous solution is the single point where they intersect. Always check your answer by substituting into both equations.

Tip — No solution means the lines are parallel; infinitely many means they are the same line.

Formula recap

{a_{1} x + b_{1} y = c_{1} a_{2} x + b_{2} y = c_{2}

A pair of linear equations.

solution = point of intersection

Where the two lines meet.

Common mistakes to avoid

Adding when you should subtract (or vice versa) during elimination.

Match the signs first: subtract for same signs, add for opposite signs.

Finding x but forgetting to find y.

A full solution gives both x and y — substitute back to get the second value.

Key takeaways

Two methods: elimination (match a coefficient, add/subtract) and substitution.
The solution is the point where the two lines intersect.
Always find both x and y, then check in both equations.

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