Algebraic Expressions: Addition and Subtraction (Part 1)

This lesson introduces algebraic expressions and shows how to add and subtract their terms. We look at what terms and coefficients are, how to tell like terms from unlike terms, and the sign rules that make combining terms reliable.

What is an algebraic expression?

An algebraic expression contains variables (letters such as $x$ and $y$) connected by addition or subtraction, unlike a purely numerical expression such as $2 + 3$ or $4 \times 6$ which contains only numbers.

$2x^2 + 3x + 5 \qquad 3x^2y^2 + 2xy + 5y \qquad \tfrac{x}{2} + \tfrac{y}{3}$

Even a single number like $7$ counts as an algebraic expression, because it can be read as $7x^0$ and $x^0 = 1$.

Terms and coefficients

An expression is made of terms joined by $+$ or $-$. The numerical coefficient is the number multiplying the variable part, and we can also state the coefficient of a particular power such as $x^2$, $x$, or $x^0$ (the constant).

For $3x^2 + 2x + 8$ the terms are $3x^2$, $2x$ and $8$.

| term | numerical coefficient | of $x^2$ | of $x$ | constant | | --- | --- | --- | --- | --- | | $3x^2$ | $3$ | $3$ | | | | $2x$ | $2$ | | $2$ | | | $8$ | $8$ | | | $8$ |

When you write out a term, keep its sign: a positive sign can be left off, but a negative sign must be written.

Like terms and unlike terms

Like terms have exactly the same variable part, so only their numbers differ.

$2x,\ -3x,\ 4x \qquad x^2y,\ -10x^2y,\ \tfrac{3}{5}x^2y \qquad 4a^2b,\ -5a^2b,\ 6a^2b$

Note that $6ba^2$ is the same as $6a^2b$, so it is a like term too. If a term has no visible number, its coefficient is $1$, not $0$.

Unlike terms have different variable parts, so they cannot be combined.

$3x,\ 4y,\ 8p \qquad xy^2,\ 8xy^2,\ 2x^2y,\ 3xy^3$

Adding like terms

To add like terms, add the coefficients and keep the variable part unchanged. Never add or change the powers.

Example. $3x^2y + 2x^2y + (-10x^2y)$

$= (3 + 2 - 10)\,x^2y = -5x^2y$

Sign rules. Positive plus positive stays positive, negative plus negative stays negative. For one positive and one negative, take the difference of the numbers and keep the sign of the larger. With several numbers, add the positives and the negatives separately, then combine.

Example. $40mn + (-10mn) + (-20mn)$. The negatives give $-10 + (-20) = -30$, then $40 + (-30) = 10$, so the result is $10mn$.

Fractions. $-\tfrac{1}{2}pq + \tfrac{3}{2}pq + (-\tfrac{5}{2}pq)$. With equal denominators, combine the numerators: $-1 + 3 - 5 = -3$, giving

$-\tfrac{3}{2}pq$

Result of zero. $20xy^2 + (-5xy^2) + (-15xy^2) = (20 - 20)xy^2 = 0$.

Subtracting like terms

To subtract a term, change the subtraction to addition, flip the sign of the second term, then add using the sign rules.

$7xy - 2xy = 5xy$

$8x^2y - (-3x^2y) = 8x^2y + 3x^2y = 11x^2y$

$-4z - (-8z) = -4z + 8z = 4z$

$-200a^2 - (-140a^2) = -200a^2 + 140a^2 = -60a^2$

$13pq^2 - (-4pq^2) = 13pq^2 + 4pq^2 = 17pq^2$

Key takeaways

• An algebraic expression is terms with variables joined by $+$ or $-$; the numerical coefficient is the number multiplying the variable part.
• Only like terms (same variable part) can be added or subtracted; unlike terms cannot.
• To combine like terms, add or subtract the coefficients and keep the variable part, applying the sign rules; to subtract, change the sign of the second term and add.