Substitution Method
Solve one equation for a variable and plug it into the other.
About this topic
Substitution solves a system by isolating one variable in one equation and replacing it in the other. This turns two equations into one equation with a single variable, which you solve, then use to find the other variable.
It is easiest when one equation already has a variable alone or with a coefficient of 1. Consider y = 2x + 1 and 3x + y = 11. Replace y in the second equation: 3x + 2x + 1 = 11, so 5x = 10 and x = 2. Back-substituting gives y = 5, so the solution is (2, 5).
The method works for nonlinear systems too, such as a line and a parabola, which is why it is so valuable in later courses. Engineers use it to solve for unknown forces, and economists use it to find where supply meets demand.
Common mistakes include substituting into the same equation you solved (which just gives a true statement like 0 = 0) and forgetting to find the second variable. If you get a false statement such as 0 = 5, the lines are parallel and there is no solution; if you get a true statement, there are infinitely many. Always check in both equations.
Key ideas
- Isolate a variable
- Substitute into the second equation
- Back-substitute to finish
Quick check
Use cases: where Substitution Method helps
- Homework: review the key idea behind Substitution Method before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Substitution Method without notes.
- Visual learning: play with the interactive lab to see how Substitution Method changes when you adjust its inputs.
- Teaching a friend: explain Substitution Method in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Substitution Method?
Solve one equation for a variable and plug it into the other.
What will I learn in Substitution Method?
Isolate a variable. Substitute into the second equation. Back-substitute to finish.
Is Substitution Method part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Substitution Method lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Substitution Method?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.