What is a system?
A system is two or more equations that must be true at the same time. For two lines, the solution is the point (x, y) that lies on both. Graphically, that is where the lines cross.
There are three possible outcomes:
- One solution: the lines cross once (different slopes).
- No solution: the lines are parallel (same slope, different intercepts). The system is inconsistent.
- Infinitely many solutions: both equations describe the same line. The system is dependent.
Substitution
Solve one equation for one variable, then substitute into the other. It is best when one variable already has coefficient 1.
Elimination
Add or subtract the equations (after scaling) so that one variable cancels. It is best when the equations are in standard form.
Worked example (substitution)
Solve 2x + 3y = 12 and x − y = 1.
- From the second equation, x = y + 1.
- Substitute: 2(y + 1) + 3y = 12, so 5y + 2 = 12 and y = 2.
- Then x = 2 + 1 = 3.
- Check in both: 2(3) + 3(2) = 12 ✓; 3 − 2 = 1 ✓.
The solution is (3, 2).
Worked example (elimination)
Solve 3x + 2y = 7 and 5x − 2y = 9.
- The y-terms are +2y and −2y, so adding the equations eliminates y: 8x = 16, so x = 2.
- Put x = 2 in the first: 6 + 2y = 7, so y = 1/2.
- Solution: (2, 1/2).
Spotting special cases
When you eliminate and get a false statement such as 0 = 7, there is no solution. When you get a true statement such as 0 = 0, there are infinitely many. For instance 2x + y = 5 and 4x + 2y = 10: doubling the first gives the second exactly, so every point on the line works.
Word problems
Define two unknowns, write two equations, solve.
Adult tickets cost $8 and child tickets $5. Ten tickets cost $62 in all. Let a be adults, c be children.
- a + c = 10
- 8a + 5c = 62
Substituting c = 10 − a gives 8a + 50 − 5a = 62, so 3a = 12 and a = 4. There are 4 adults and 6 children: 32 + 30 = 62. ✓
Common mistakes
- Solving for only one variable and stopping.
- Not multiplying every term when scaling an equation.
- Not checking the answer in both original equations.