Graphing is hard to read exactly when the answer involves fractions. Substitution finds the exact answer with algebra, using what you know about equations with variables on both sides.
Why it works
If one equation says $y = 2x - 1$, then at the solution $y$ and $2x - 1$ are the same number. So in the other equation you may replace $y$ with $2x - 1$. That leaves one equation with one unknown, which you already know how to solve.
Key idea
(1) Solve one equation for one variable, if it isn’t already. (2) Substitute that expression into the other equation. (3) Solve. (4) Put the value back to find the other variable. (5) Check the pair in both equations.
Worked examples
Worked example: One equation is solved for y
Solve: $y = 2x - 1$ and $3x + y = 9$.
- Replace $y$ in the second equation: $3x + (2x - 1) = 9$.
- Combine: $5x - 1 = 9$, so $5x = 10$ and $x = 2$.
- Then $y = 2 \cdot 2 - 1 = 3$. Check: $3 \cdot 2 + 3 = 9$.
Answer: $(2, 3)$
Worked example: Both are solved for y
Solve: $y = 4x - 3$ and $y = x + 6$.
- Both expressions equal $y$, so they equal each other: $4x - 3 = x + 6$.
- Subtract $x$ and add 3: $3x = 9$, so $x = 3$.
- $y = 3 + 6 = 9$. Check: $4 \cdot 3 - 3 = 9$.
Answer: $(3, 9)$
Worked example: Solve for a variable first
Solve: $x + 2y = 7$ and $3x - y = 7$.
- The first equation has $x$ with coefficient 1: $x = 7 - 2y$.
- Substitute: $3(7 - 2y) - y = 7$, so $21 - 7y = 7$.
- $-7y = -14$, so $y = 2$. Then $x = 7 - 4 = 3$.
- Check: $3 + 4 = 7$ and $9 - 2 = 7$.
Answer: $(3, 2)$
Common mistake
Substituting back into the same equation the expression came from. Putting $y = 2x - 1$ into $y = 2x - 1$ gives $2x - 1 = 2x - 1$, which is always true and tells you nothing. Substitute into the other equation.