Algebra
Systems of equations
A system asks for the values that satisfy two or more equations at once. Graphically, that is the point where the lines cross. Algebraically, you can substitute one equation into the other or eliminate a variable by adding.
Why it matters
Systems appear in test prep, in physics and economics modelling, and again in linear algebra. Choosing the efficient method — rather than always substituting — is what separates a two-minute answer from a six-minute one.
Worked example
Solve 2x + 3y = 12 and 4x − 3y = 6
- 1Notice the y terms are opposites. +3y and −3y cancel when added
- 2Add the equations. 6x = 18
- 3Solve for x. x = 3
- 4Back-substitute. 2(3) + 3y = 12 → 3y = 6 → y = 2
Answer: (x, y) = (3, 2)
Common mistakes
- Adding equations when the coefficients are not opposites.
- Solving for one variable and forgetting to find the other.
- Not checking the answer in both equations.
How to check your answer
- Substitute the pair into both original equations.
- If both variables vanish: a true statement means infinitely many solutions, a false one means no solution.
Related topics
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