Systems of Linear Equations
“The solution is the one point where two lines cross.”
The formula
a₁x + b₁y = c₁, a₂x + b₂y = c₂How to read it: find the one pair (x, y) satisfying both linear equations at once
- x, y
- — the two unknowns to find together
- a₁, b₁, a₂, b₂
- — the coefficients in each equation
- (x, y)
- — the crossing point of the two lines = the solution
The hook
With two unknowns, one equation isn't enough — it's a game of finding the single spot where two conditions meet.
In plain words
Finding the one pair (x, y) that satisfies both unknowns at once. You need two equations to pin the answer to a single point.
The intuition
Each linear equation is a straight line on the plane. The coordinates where the two lines cross are exactly the solution — the one place that meets both conditions together.
How it's built
Two routes. Elimination: add or subtract the equations to wipe out one unknown. Substitution: rearrange one equation to y=… and drop it whole into the other. Either way, the key is shrinking to a single unknown.
Example
x+y=5, x−y=1. Add them: 2x=6 → x=3. Back-substitute: y=2. The pair (3,2) makes both equations true.
Common mistake
There isn't always exactly one solution — if the two lines are parallel they never meet (no solution), and if they fully overlap there are infinitely many.
Where it's used
Any problem where two things are tangled — adult vs. child tickets, prices of two items, speed and time — lets you tease out each one.
Where it came from
The idea of eliminating unknowns to solve appears 2000 years ago in China's 'Nine Chapters', already laid out in a matrix-like table.
Prerequisites
Quick check
For x + y = 10 and x − y = 2, what is x?
- 4
- 6✓
- 8
- 10
Practice
For x + y = 7 and x − y = 3, find x.
Answer: 5
- Add the two equations to cancel y: 2x = 10
- x = 5
Takeaway: Elimination: add or subtract to wipe out one unknown.
For y = 2x and x + y = 9, find y.
Answer: 6
- Substitute: x + 2x = 9 → 3x = 9 → x = 3
- y = 2·3 = 6
Takeaway: Substitution: drop one unknown whole into the other equation.
Solve 2x + y = 7 and x + y = 4, then check your answer in both equations.
Answer: undefined
- Subtract the lower from the upper: (2x+y) − (x+y) = 7 − 4 → x = 3
- Substitute into x + y = 4: y = 1
- Check: 2·3+1 = 7 ✓, 3+1 = 4 ✓
Takeaway: The (x, y) you find must satisfy 'both' equations.