Simultaneous Equations: Substitution, Elimination + Examples

Simultaneous linear equations are two equations, each containing two variables, that must be solved together. For example x + y = 10 and 2x − y = 2. The solution is the pair of values (x, y) that makes both equations true at once. This is taught in Year 8 (SMP).

Why Two Equations Are Needed

A single equation in two variables such as x + y = 10 has infinitely many answers: (1, 9), (2, 8), (3, 7) and so on. Only when a second equation is added does the answer narrow to one pair. Graphically each equation is a straight line, and the solution is the point where the two lines cross.

General form:

Method 1: Substitution

Use this when one variable is already isolated or easy to isolate.

Example: x + y = 10 and 2x − y = 2

1. From the first equation: y = 10 − x
2. Substitute into the second: 2x − (10 − x) = 2
3. Expand: 2x − 10 + x = 2 → 3x = 12 → x = 4
4. Back-substitute: y = 10 − 4 = 6
5. Check: 4 + 6 = 10 ✔ and 2(4) − 6 = 2 ✔

The solution is (4, 6).

Method 2: Elimination

Use this when the coefficients of one variable are already equal or opposite.

Same example:

Because +y and −y cancel, we get 3x = 12 → x = 4, then y = 6.

If the coefficients do not match yet, multiply one equation first. For 3x + 2y = 16 and 5x − 2y = 8 the y coefficients are already opposite, so simply add.

Method 3: Mixed

Eliminate to get one variable, then substitute to get the other. This is the method most used in exams because it takes the fewest arithmetic steps.

Which Method to Pick

A Worked Word Problem

Two books and three pencils cost Rp 21,000. Four books and one pencil cost Rp 27,000. What does one book cost?

Let the price of a book be x and a pencil y.

• 2x + 3y = 21,000
• 4x + y = 27,000

Multiply the second equation by 3 → 12x + 3y = 81,000. Subtract the first → 10x = 60,000 → x = 6,000. Then y = 27,000 − 4(6,000) = 3,000.

One book costs Rp 6,000 and one pencil Rp 3,000.

The Mistakes That Actually Cost Marks

1. Sign errors when subtracting one equation from another. This is the number one cause, and the root is not simultaneous equations but integer arithmetic. A child who hesitates over 8 − (−3) will stumble at the elimination step every time.

2. Forgetting to multiply every term. When multiplying 4x + y = 27,000 by 3, the right-hand side must be multiplied too.

3. Stopping after finding one variable. The answer is always a pair (x, y), never a single number.

4. Checking only one equation. A correct solution must satisfy the first and the second equation. Checking just one lets wrong answers through.

5. Mislabelling in word problems. If x and y are swapped, every calculation is correct and the answer is still wrong. Write "x = price of a book" explicitly before calculating anything.