Algebra
Systems of equations and inequalities
Solving systems by substitution
20 practice questions
2 video lessons
Theory + worked examples
Theory
Substitution solves a system in four steps:
- Solve one equation for a variable.
- Substitute that expression into the other equation.
- Solve for the remaining variable.
- Back-substitute to find the first variable.
Best when a variable is already isolated (like \(y=\ldots\)).
The substitution steps.
A worked example.
The idea:
\[y=f(x)\ \text{into}\ (\text{other equation})\]
You get one equation in one variable to solve.
How to solve by substitution
- Isolate a variable in one equation.
- Substitute into the other.
- Solve the single-variable equation.
- Back-substitute and check.
Example 1 — Direct substitution
Solve \(y=2x\) and \(x+y=6\).
Solution
Substitute \(2x\) for \(y\).
| \(x+2x\) | \(=\) | \(6\) |
| \(x\) | \(=\) | \(2\) |
| \(y\) | \(=\) | \(4\) |
Example 2 — Solve for a variable first
Solve \(x-y=1\) and \(2x+y=8\).
Solution
Solve the first for \(x=y+1\), then substitute.
| \(2(y+1)+y\) | \(=\) | \(8\) |
| \(y\) | \(=\) | \(2\) |
| \(x\) | \(=\) | \(3\) |
Example 3 — Back-substitute
After finding \(x=2\) with \(y=2x\), find \(y\).
Solution
Substitute back.
| \(y\) | \(=\) | \(2(2)=4\) |
Example 4 — When to use it
When is substitution easiest?
Solution
When one equation is already solved for a variable (like \(y=\dots\)).
Common pitfalls
Substitute into the OTHER equation, not the same one.
Back-substitute to find both variables.
Use parentheses when substituting an expression.
Frequently asked questions
What is substitution?
Replacing one variable with an expression from the other equation.
When is substitution easiest?
When a variable is already solved for.
What do you do after solving for one variable?
Back-substitute to find the other.
Do you substitute into the same equation?
No — into the other one.
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