Lesson 2: Adding and Subtracting Linear Expressions

Property

To add like terms: Add the numerical coefficients of the terms. Do not change the variable factors of the terms.
To subtract like terms: Subtract the numerical coefficients of the terms. Do not change the variable factors of the terms.
Replacing an expression by a simpler equivalent one is called simplifying the expression.

Examples

• To simplify $7b + 4b$, we add the coefficients: $(7+4)b = 11b$. We have combined the like terms into a single, simpler term.
• To simplify $12k - (-3k)$, we subtract the coefficients: $(12 - (-3))k = (12+3)k = 15k$. The variable factor $k$ remains unchanged.
• We cannot simplify $8p + 5q$ because $8p$ and $5q$ are not like terms. Their variable factors are different.

Explanation

When combining like terms, you only perform the operation on their coefficients. The variable part just tells you what 'family' of terms you're counting. It's like adding 4 apples and 5 apples to get 9 apples.

Property

To add linear expressions, we remove parentheses and combine like terms.
Like terms have the same variable with the same exponent.
For linear expressions, we combine constant terms together and variable terms with the same variable together.
To add like terms, we add their numerical coefficients.

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.