College Algebra
Systems of Linear Equations
Lesson
A system of linear equations is a set of equations that share the same variables. A solution is a set of values that makes every equation in the system true at once. With two variables, the solution is an ordered pair ; with three, it’s an ordered triple .
Two standard methods:
- Substitution: solve one equation for one variable, then plug that expression into the other equation. Best when one variable is already isolated (or close to it).
- Elimination: add or subtract equations (multiplied by suitable constants if needed) to cancel a variable. Best when coefficients line up cleanly.
Worked example 1 — elimination
Add the equations:
Plug back: , so .
Worked example 2 — substitution
From the second equation, . Substitute:
Then .
For three-variable systems, use the same tools repeatedly: eliminate one variable to reduce to a two-variable system, solve that, then back-substitute.
How to type your answer
Type the values in order, separated by commas — no parentheses. For two-variable systems: first, then . For three-variable: , , . Examples: 7,3, -1,3, 1,2,3.
Practice
Work through these. Stuck? Click Get a hint.
Warm-Up
Quick problems to get going.
Problem 1
Problem 2
Problem 3
Problem 4
Practice
Standard problems matching the lesson.
Problem 5
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Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Challenge
Harder problems — edge cases, trickier numbers, multiple steps.
Problem 15
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Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22