Simultaneous Equations

Simultaneous equations

Simultaneous equations are two equations with two unknowns (usually x and y) that are true at the same time. Solving them finds the pair of values that satisfies both. There are two main methods: elimination and substitution.

Method 1: Elimination

The idea: add or subtract the equations to eliminate one unknown.

Example: solve

3x + y = 11   ...(1)
x + y = 5     ...(2)

1. The y terms match, so subtract (2) from (1): 2x = 6 → x = 3.

2. Substitute x = 3 into (2): 3 + y = 5 → y = 2.

• Solution: x = 3, y = 2.

If the coefficients don't match, multiply one (or both) equations first so they do. Same signs → subtract; different signs → add.

Example:

2x + 3y = 13
4x − y = 5

Multiply the second by 3: 12x − 3y = 15. Now add to the first (3y and −3y cancel): 14x = 28 → x = 2, then y = 3.

Method 2: Substitution

Rearrange one equation to make a letter the subject, then substitute into the other.

Example:

y = 2x + 1
3x + y = 11

Substitute the first into the second: 3x + (2x + 1) = 11 → 5x + 1 = 11 → x = 2, then y = 2(2)+1 = 5.

Graphical meaning

The solution is the point where the two lines cross on a graph. Two parallel lines (no crossing) mean no solution.

Worked example

Solve x + y = 7 and x − y = 1 by elimination.

1. Add the equations (y and −y cancel): 2x = 8 → x = 4.

2. Substitute into x + y = 7: 4 + y = 7 → y = 3.

• Solution: x = 4, y = 3. ✓

Common mistakes

• Adding when you should subtract (or vice versa) — match the signs of the term you're eliminating.
• Only finding one unknown — always substitute back to get the second.
• Forgetting to multiply every term when scaling an equation.

Exam tips

• Label your equations (1) and (2) and show each step.
• After finding both values, check them in the other equation.
• Choose the easier method: elimination if coefficients match, substitution if one is already y = ….

Key facts to remember

• Simultaneous equations = two equations, two unknowns, true together; the solution is where the lines cross.
• Elimination: make one variable's coefficients equal, then add (different signs) or subtract (same signs).
• Substitution: make one letter the subject and put it into the other; always find both values and check.