Solving two at once
Simultaneous linear equations are two equations in x and y that must both be true at the same time. Their solution is the single pair (x, y) that fits both. We use two methods: substitution and elimination.
Substitution method
Worked example / Worked example
Solve y = x + 1 and x + y = 7.
Substitute y = x + 1 into the second: x + (x + 1) = 7.
So 2x + 1 = 7, giving 2x = 6 and x = 3.
Then y = 3 + 1 = 4. Solution: (3, 4).
Elimination method
When coefficients match, add or subtract the equations to remove one variable.
Key idea / Key idea
If a variable has the same coefficient in both equations, subtract the equations to eliminate it; if the signs are opposite, add them.
Worked example / Worked example
Solve x + y = 8 and x − y = 2.
Add the equations: 2x = 10, so x = 5.
From x + y = 8: 5 + y = 8, so y = 3. Solution: (5, 3).
Remember / Remember
- The answer is one pair (x, y) fitting both equations.
- Substitution: put one equation into the other.
- Elimination: add or subtract to cancel a variable.
- Check the pair in both equations.