Lesson 7: Subtract Expressions

Property

When distributing a negative sign across parentheses: $-(a + b) = -a - b$ and $-(a - b) = -a + b$.

When multiplying by a negative coefficient: $-c(a + b) = -ca - cb$ and $-c(a - b) = -ca + cb$.

Property

When subtracting linear expressions, you can rewrite subtraction as adding the opposite (additive inverse).
For any linear expressions $A$ and $B$, we have $A - B = A + (-B)$. The opposite of an expression changes the sign of every term.

Examples

Property

When a variable in both equations has the exact same coefficient (e.g., $4x$ and $4x$), adding the equations will not eliminate it. Instead, you must subtract the entire second equation from the first.

To do this correctly, you must distribute the negative sign to every single term in the bottom equation (changing all their signs) and then add the equations together.

Examples

• Distributing the Negative: Subtract $(4x - 7)$ from $(6x + 2)$.

Write it out: $(6x + 2) - (4x - 7)$.
Distribute the minus sign to flip the signs inside: $6x + 2 - 4x + 7$.
Combine like terms: $2x + 9$.

• Subtracting Equations: Solve $5x + 3y = 17$ and $2x + 3y = 8$.

Since the $y$ terms are identical ($3y$), subtract the entire bottom equation:
$-(2x + 3y = 8) \rightarrow -2x - 3y = -8$
Now add this to the top equation:
$(5x - 2x) + (3y - 3y) = 17 - 8$
$3x = 9 \rightarrow x = 3$.
Back-substitute: $2(3) + 3y = 8 \rightarrow 6 + 3y = 8 \rightarrow 3y = 2 \rightarrow y = \frac{2}{3}$.

Explanation

Subtracting an entire equation is exactly the same as multiplying it by -1 and then adding. The most common mistake in algebra is subtracting the first term but forgetting to subtract the rest! To avoid this trap, do not try to subtract in your head. Physically draw parentheses around the bottom equation, write a minus sign outside, and rewrite the equation with every single sign flipped. Then, just add them normally.