Math Core

An expression like $2g + 15$ describes a phone bill for any amount of data. But when the bill arrives, you want an actual number. Finding that number is called evaluating the expression.

Substitute, then simplify

How to evaluate an expression

Substitute: replace each variable with its value. Put the value in parentheses so you remember to multiply.
Simplify: follow the order of operations.

For example, to evaluate $2g + 15$ when $g = 8$ , write $2(8) + 15$ . Multiply first to get $16 + 15 = 31$ . The bill is $31.

Common mistake

When a number replaces a variable next to a coefficient, you must multiply, not just write the digits side by side. If $g = 8$ , then $2g$ is $2 \times 8 = 16$ , not 28. Writing $2(8)$ with parentheses keeps you from making this mistake.

Worked example: One variable, with an exponent

Evaluate $3x^2 + 4$ when $x = 5$ .

\begin{aligned} 3x^2 + 4 &= 3(5)^2 + 4 && \text{substitute} \\ &= 3(25) + 4 && \text{exponent first} \\ &= 75 + 4 && \text{multiply} \\ &= 79 && \text{add} \end{aligned}

The exponent belongs to $x$ only, so you square 5 before multiplying by 3.

Worked example: Two variables

Evaluate $\dfrac{a + b}{4} + ab$ when $a = 9$ and $b = 3$ .

\begin{aligned} \dfrac{a + b}{4} + ab &= \dfrac{9 + 3}{4} + (9)(3) && \text{substitute} \\ &= \dfrac{12}{4} + 27 && \text{top of the fraction; multiply} \\ &= 3 + 27 && \text{divide} \\ &= 30 && \text{add} \end{aligned}

Fractions and decimals

Variables can stand for fractions and decimals, too. The steps are exactly the same.

Worked example: A fraction value

Evaluate $6y - 1$ when $y = \dfrac{2}{3}$ .

6y - 1 = 6\left(\dfrac{2}{3}\right) - 1 = \dfrac{12}{3} - 1 = 4 - 1 = 3

Formulas

A formula is an expression (or equation) that gives a rule for finding a quantity. Many formulas come from geometry. Evaluating a formula works just like evaluating any expression.

quantity	formula
area of a rectangle	$A = \ell w$ (length times width)
perimeter of a rectangle	$P = 2\ell + 2w$
area of a square	$A = s^2$
volume of a cube	$V = s^3$
distance traveled	$d = rt$ (rate times time)

Worked example: Perimeter of a rectangle

A garden is $12.5$ meters long and $8$ meters wide. Use $P = 2\ell + 2w$ to find its perimeter.

P = 2(12.5) + 2(8) = 25 + 16 = 41

The perimeter is 41 meters.

A rectangle 12.5 m long and 8 m wide.

Tip

After you evaluate, ask yourself if the answer makes sense. A garden that is about 12 m by 8 m should have a perimeter a bit more than $2(12) + 2(8) = 40$ meters. An answer of 41 fits.