Solving Systems by Substitution
Systems of Linear Equations
A system of equations is a set of two or more equations with the same variables. Solving a system means finding the values that satisfy all equations simultaneously — graphically, this is the point where the lines intersect.
The substitution method solves one equation for one variable and then substitutes that expression into the other equation, reducing the system to a single equation in one unknown.
This lesson teaches you to apply substitution step by step, identify special cases (no solution and infinitely many solutions), and verify your answer by plugging it back into both original equations.
A system of two linear equations has two equations and two unknowns. The solution is the point $(x, y)$ satisfying both.
Classification
| Type | Graph | Solutions |
|---|---|---|
| Consistent & Independent | Lines intersect | One solution |
| Inconsistent | Parallel lines | No solution |
| Dependent | Same line | Infinitely many |
Substitution Method
- Solve one equation for one variable.
- Substitute into the other equation.
- Solve the resulting one-variable equation.
- Back-substitute to find the other variable.
$y = 2x + 1$ and $3x + y = 11$
- Substitute: $3x + (2x + 1) = 11$
- $5x + 1 = 11 \;\Rightarrow\; x = 2$
- $y = 2(2) + 1 = 5$
Solution: $(2, 5)$. Check: $3(2) + 5 = 11$ ✓
$x + 2y = 7$ and $3x - y = 7$
From eq 1: $x = 7 - 2y$. Substitute: $3(7 - 2y) - y = 7$.
$21 - 6y - y = 7 \;\Rightarrow\; -7y = -14 \;\Rightarrow\; y = 2$, $x = 3$.
Solution: $(3, 2)$.
Practice Problems
Show Answer Key
1. $(3, 6)$
2. $(12, 3)$
3. $(3, -1)$
4. $(5, 3)$
5. $(4, 2)$
6. No solution (parallel, inconsistent)
7. $(\dfrac{7}{5}, \dfrac{8}{5})$
8. Infinitely many (dependent — same line)
9. $(3, 2)$
10. $(2, 1)$