Substitution, elimination, and how to spot when a system has one, none, or infinitely many solutions.
Concept
A system of two linear equations in two variables means: find an (x, y) pair that satisfies both equations at once. Geometrically, it's the point where two lines intersect.
Substitution. Solve one equation for a single variable, then plug that expression into the other.
Elimination. Multiply one or both equations so that the coefficients of one variable cancel when you add or subtract.
How many solutions a system has comes straight from the slopes:
One solution — lines have different slopes (they cross once).
No solution — lines are parallel: same slope, different y-intercepts.
Infinitely many solutions — lines are the same: equal slopes and equal y-intercepts (one equation is a multiple of the other).
Worked example 1 · substitution
Solve y = 2x + 1 and 3x + y = 11.
Solution
Step 1. The first equation is already solved for y. Substitute 2x + 1 for y in the second: 3x + (2x + 1) = 11
Step 2. Combine and solve: 5x + 1 = 11 ⇒ 5x = 10 ⇒ x = 2
Step 3. Back-substitute: y = 2(2) + 1 = 5
Solution: (2, 5). Check: 3(2) + 5 = 11. ✓
Worked example 2 · elimination
Solve 2x + 3y = 12 and 4x − 3y = 6.
Solution
Step 1. The +3y and −3y already cancel. Add the two equations: 6x = 18
Step 2. Solve: x = 3
Step 3. Plug back into the first: 2(3) + 3y = 12 ⇒ 3y = 6 ⇒ y = 2
Solution: (3, 2). Check the second equation: 4(3) − 3(2) = 12 − 6 = 6. ✓
Practice test
8 questions on substitution, elimination, and recognizing one / none / infinite solutions.
Practice test · 8 questionsQuestion 1 of 8 · Score 0