Solving Two-Variable Systems
Graphing shows a system's solution but rarely gives exact values. Substitution and elimination find the solution algebraically, no graph required.
Substitution shines when one variable is already isolated; elimination shines when adding the equations cancels a variable. Both reach the same answer.
Substitution
Solve one equation for a variable, then substitute that expression into the other. Now you have a single equation in one variable to solve.
Back-substitute the value you found to get the other variable. This works best when a coefficient is already .
Whichever method you use, the answer is the single point that lies on both lines — the crossing point shown below for a sample system. Substitution and elimination just find its coordinates exactly, without drawing.
Elimination
Line the equations up and add or subtract so one variable cancels. You may need to multiply an equation first to make coefficients match.
After one variable is eliminated, solve for the other and back-substitute. If both variables vanish and you get a true statement, there are infinitely many solutions; a false statement means none.
Worked examples
Example 1: elimination
Solve and .
Answer:
Example 2: substitution
Solve and .
Answer:
Example 3: elimination after multiplying
Solve and .
Answer:
Try one yourself
Common questions
Substitution or elimination — which should I use?
Use substitution when a variable is already isolated (or has coefficient 1). Use elimination when adding or subtracting the equations neatly cancels a variable.
What if both variables cancel?
A true statement (like ) means infinitely many solutions; a false one (like ) means no solution.
Do the two methods ever give different answers?
No. A consistent system has one solution set, and both methods find it — pick whichever is less work.
Want the video version?
Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.