MathematicsExam code: 0580
Simultaneous Equations
Algebra and Sequences
When one unknown is already isolated
- Rearrange one equation to make an unknown the subject, then substitute that expression into the other
- This suits an equation such as 2x − y = 1, which rearranges easily to y = 2x − 1
- Substituting gives one equation in one unknown
- With 4x + 3y = 27, substituting y = 2x − 1 gives 4x + 3(2x − 1) = 27, so 10x = 30 and x = 3, giving y = 5
- Substitution is the method that also handles non-linear pairs
Worked example
Solving by substitution
Solve the simultaneous equations y = 2x + 1 and 3x + y = 11.
Solution:
- One equation already has y on its own, so substitute that expression into the other
- 3x + (2x + 1) = 11
- Collect: 5x + 1 = 11
- Subtract 1: 5x = 10, so x = 2
- Substitute back into the isolated equation: y = 2 × 2 + 1 = 5
- The solution is x = 2, y = 5
- Check in the equation not used for the substitution: 3 × 2 + 5 = 11