Two variables together
A linear equation in two variables has two unknowns, usually x and y, each to the first power. Examples are x + y = 5 and 2x − y = 3. Instead of one answer, such an equation has many pairs of values (x, y) that make it true.
Finding pairs of solutions
Choose a value for one variable, then work out the other. This gives an ordered pair (x, y).
Worked example / Worked example
Find three solutions of x + y = 6.
If x = 0, then y = 6 → (0, 6).
If x = 2, then y = 4 → (2, 4).
If x = 5, then y = 1 → (5, 1).
Each pair makes x + y = 6 true.
Key idea / Key idea
A solution of an equation in two variables is an ordered pair (x, y). Substitute both values; the equation must balance.
Checking a pair
To test whether (x, y) is a solution, put both numbers into the equation. For 2x − y = 3, try (3, 3): 2 × 3 − 3 = 6 − 3 = 3. True, so (3, 3) is a solution. Try (1, 2): 2 × 1 − 2 = 0 ≠ 3, so (1, 2) is not a solution.
Remember / Remember
- Two variables → many solution pairs.
- A solution is an ordered pair (x, y).
- Substitute both values to check.