Add and Subtract Algebraic Expressions

This lesson covers how to add and subtract algebraic expressions. The key idea is to collect like terms, terms with the same variables raised to the same powers, and combine their coefficients. There are two layouts: the column method, where you stack the expressions and line up like terms, and the row method, where you write everything in a single line and group the like terms together.

Like terms

An expression is made of terms joined by addition or subtraction. Two terms are like terms when they have exactly the same variable part, for example $3x^{2}$ and $4x^{2}$, or $2x$ and $2x$. Only like terms can be added or subtracted. Constants such as $-5$ and $-8$ are like terms with each other.

Example 1: add $3x^{2}+2x-5$ and $4x^{2}+2x-8$

Column method. Write the second expression under the first so that like terms line up, then add each column.

$\begin{aligned} &3x^{2}+2x-5 \\ +\;&4x^{2}+2x-8 \end{aligned}$

$x^{2}$ terms: $3+4 = 7$, giving $7x^{2}$. $x$ terms: $2+2 = 4$, giving $4x$. Constants: $-5+(-8) = -13$.

$3x^{2}+2x-5 \;+\; 4x^{2}+2x-8 = 7x^{2}+4x-13$

Row method. Write both in a line and group the like terms:

$(3x^{2}+4x^{2})+(2x+2x)+(-5-8) = 7x^{2}+4x-13$

Both methods give the same result.

Example 2: add $4xy-8x^{2}y+4x-10$ and $-7xy+8x^{2}y+4x+10$

Group each set of like terms:

$(4xy-7xy)+(-8x^{2}y+8x^{2}y)+(4x+4x)+(-10+10)$

$xy$ terms: $4+(-7) = -3$, giving $-3xy$. $x^{2}y$ terms: $-8+8 = 0$, so this term disappears. $x$ terms: $4+4 = 8$, giving $8x$. Constants: $-10+10 = 0$.

$= -3xy+8x$

Example 3: add $8x^{2}+7x-8$, $\;3x^{2}+4x+4$, and $-2x^{2}+4x-10$

With three expressions, stack all three and add each column.

$x^{2}$ terms: $8+3+(-2) = 9$. $x$ terms: $7+4+4 = 15$. Constants: $-8+4+(-10) = -14$.

$8x^{2}+7x-8 \;+\; 3x^{2}+4x+4 \;+\; -2x^{2}+4x-10 = 9x^{2}+15x-14$

When a column mixes positive and negative numbers, take the difference and keep the sign of the larger one. When they share a sign, just add and keep that sign.

Example 4: add expressions in $a^{2}b^{2}$, $ab^{2}$ and $ab$

Line up like terms across the expressions, writing a zero coefficient where a term is missing, then add each column. Combining the like terms gives:

$a^{2}b^{2}$ terms: $10+7 = 17$. $ab^{2}$ terms: $4+9-8 = 5$. $ab$ terms: $-8-6-7+3 = -18$. Constants: the constant column totals $13$.

$= 17a^{2}b^{2}+5ab^{2}-18ab+13$

Example 5: find the sum of $2a^{2}b+6ab^{2}+7$ and $-8ab^{2}+4a^{2}b+4$

Collect like terms:

$a^{2}b$ terms: $2+4 = 6$. $ab^{2}$ terms: $6+(-8) = -2$ (take the difference, keep the sign of the larger). Constants: $7+4 = 11$.

$= 6a^{2}b-2ab^{2}+11$

Subtracting expressions

To subtract one expression from another, change the subtraction to addition and flip the sign of every term in the second expression. In other words, add the inverse of the second expression.

Example 6: $\,(2x^{2}+4x-9)-(3x^{2}+8x+10)$

Flip the signs of the second expression and add:

$2x^{2}+4x-9 \;+\; (-3x^{2}-8x-10)$

$x^{2}$ terms: $2-3 = -1$. $x$ terms: $4-8 = -4$. Constants: $-9-10 = -19$.

$= -x^{2}-4x-19$

Example 7: $\,(2x^{2}-2x)-(-3x^{2}+4x)$

Flip the signs of the second expression and add. The $x^{2}$ terms give $2+3 = 5$, so $5x^{2}$. The $x$ terms give $-2-4 = -6$, so $-6x$.

$= 5x^{2}-6x$

Example 8: subtract $4x^{2}+10x+5$ from $3x^{2}-8x+10$

Write the expression you subtract from first, then flip the signs of the one being subtracted:

$3x^{2}-8x+10 \;+\; (-4x^{2}-10x-5)$

$x^{2}$ terms: $3-4 = -1$. $x$ terms: $-8-10 = -18$. Constants: $10-5 = 5$.

$= -x^{2}-18x+5$

Key takeaways

• Only like terms, those with the same variables and powers, can be added or subtracted; combine their coefficients with the correct signs.
• The column method and the row method give the same answer; use whichever layout is clearer.
• To subtract an expression, change the sign of every term in it and add, that is, add the inverse.