Two phone plans crossing
One plan costs $10 plus $1 a gigabyte, another costs $20 with no per-gigabyte fee. Draw both as lines and they cross at the usage where the plans cost the same — that crossing point is the solution to the system of equations.
Two equations, one shared point
A system of equations is two or more equations considered together. Its solution is the pair (x, y) that makes every equation true at once — graphically, the point where the lines cross.
Substitution
Solve one equation for a variable, then substitute that expression into the other. This turns two equations into one you can solve directly.
y = x + 1 and y = 2x − 1. Set them equal: x + 1 = 2x − 1 → x = 2, so y = 3. The lines cross at (2, 3).
Simulation · Line grapher: y = mx + b
y = 1x + 2
Graph two lines and see that the solution is exactly where they intersect.
Lab · Solve by substitution
- Take y = x + 4 and y = 3x.
- Set the two right-hand sides equal to each other.
- Solve for x, then find y.
What you should see: x + 4 = 3x → 4 = 2x → x = 2, and y = 3(2) = 6. Solution: (2, 6).