Systems of linear equations
The idea
One equation with two unknowns leaves infinitely many possibilities, but a second equation about the same unknowns can pin them down. A system of linear equations is a pair of conditions that must hold at the same time, and its solution is the pair of values satisfying both. Graphically, each equation draws a line of its own possibilities, and the solution is where the two lines cross — the single point that lives on both. You already solve one-variable equations, and systems reduce to exactly that.
Substitution is the workhorse: solve one equation for one unknown, then substitute that expression into the other equation, leaving a single equation in a single variable. The trap is treating the equations separately — finding a pair that fits one equation and declaring victory. A genuine solution must check out in both equations, so always substitute your final pair into the equation you did not just use. If it fails there, the work has a slip somewhere.
Worked example
A movie theater sells adult tickets for $8 and child tickets for $5. A family buys 7 tickets for a total of $44. How many of each type did they buy?
- Name the unknowns and translate both facts: with a adult and c child tickets, the count gives a + c = 7 and the money gives 8a + 5c = 44.
- Solve the simpler equation for one unknown: a = 7 − c, then substitute into the money equation: 8(7 − c) + 5c = 44.
- Distribute and combine like terms: 56 − 8c + 5c = 44, so 56 − 3c = 44, which gives 3c = 12 and c = 4.
- Back-substitute to finish: a = 7 − 4 = 3, then verify in the money equation: 8 × 3 + 5 × 4 = 24 + 20 = 44, so the pair satisfies both conditions.
Answer. The family bought 3 adult tickets and 4 child tickets.
Check your understanding
- Why does it take two equations to pin down two unknowns, and what does one equation alone leave open?
- What does the solution of a system look like on a graph, and what would parallel lines mean for the answer?
- How do you choose which equation and which variable to use for substitution to keep the algebra light?
- Why is checking the final pair in both original equations a real test rather than a formality?