Systems of Equations & Inequalities

Systems by Elimination

Solving a system of linear equations using elimination.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Systems by Substitution

Canceling a Variable by Combining Equations

Neither 4x + 3y = 18 nor 5x − 3y = 9 has a variable conveniently isolated — substitution here would drag fractions into every step. Before reading on, notice something about the y-coefficients, 3 and −3: what would happen if you added these two equations together, left side to left side and right side to right side?

Definition — Elimination

Solving a system by adding or subtracting the two equations (after possibly scaling one or both by a constant), chosen so that one variable's coefficients cancel exactly, leaving a single equation in one unknown.

Elimination is valid because of two basic properties of equality: multiplying both sides of a true equation by the same nonzero number keeps it true, and adding the left and right sides of two true equations together produces another true equation. Combining those two facts means any scaled sum of two true equations is still guaranteed true — a powerful tool once one variable's coefficients are engineered to cancel.

Worked Example — Eliminating Directly

Solve 4x + 3y = 18 and 5x − 3y = 9. The y-coefficients, 3 and −3, are already opposites. Add the equations directly: 9x = 27, so x = 3. Substitute into either original equation: 4(3) + 3y = 18, giving 3y = 6, so y = 2. The solution is (3, 2).

Worked Example — Scaling One Equation First

Solve 2x + y = 11 and x + 3y = 18. Neither variable's coefficients match or are opposite yet. Multiply the first equation by −3 to make the y-coefficients opposite: −6x − 3y = −33. Add this to the second equation: −5x = −15, so x = 3. Substitute into 2x + y = 11: 6 + y = 11, giving y = 5. The solution is (3, 5).

Worked Example — Scaling Both Equations

Solve 3x + 4y = 10 and 5x + 6y = 16. To eliminate x, find a common multiple of 3 and 5: multiply the first equation by 5 and the second by −3: 15x + 20y = 50, and −15x − 18y = −48. Adding: 2y = 2, so y = 1. Substitute into the first original equation: 3x + 4 = 10, giving x = 2. The solution is (2, 1).

Tip

Before scaling anything, scan both equations for a variable whose coefficients are already equal or opposite — that variable eliminates immediately, with no extra scaling step needed at all.

Common Mistakes

Adding two equations when the target variable's coefficients are equal (not opposite), causing that variable to double instead of cancel.
Check whether the coefficients being eliminated are opposites (then add the equations) or equal with the same sign (then subtract the equations) before combining them.
Scaling only one side of an equation, or scaling only one equation's variable term without scaling its constant term too.
Multiplying an equation by a constant means multiplying every single term on both sides — the constant term and both variable terms all get scaled together.

Key Takeaways

Elimination adds or subtracts two equations, after scaling if needed, to cancel one variable entirely.
This works because scaling a true equation and adding two true equations both preserve truth.
Scanning for already-matching or opposite coefficients before scaling saves an unnecessary step.

Summary

Elimination handles systems that would otherwise force messy fractions on substitution. The final lesson in this unit extends systems thinking from equations to inequalities, where the solution becomes a shaded region instead of a single point.

Track your progress

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