Lesson 16: Simplifying and Evaluating Variable Expressions

New Concept

To evaluate an expression that contains variables, substitute each variable in the expression with a given numeric value, and then find the value of the expression.

What’s next

This lesson is your first step. Next, you'll apply this core concept by working through examples with exponents, simplification, and real-world application problems.

Property

To evaluate an expression that contains variables, substitute each variable in the expression with a given numeric value, and then find the value of the expression.

Examples

Evaluate $(−b+c) − (b − c)$ for $b = 5$ and $c = -3$: $[-5+(-3)] - [5-(-3)] = -8 - 8 = -16$.
Evaluate $(ab)(cab)$ for $a=3, b=-2, c=1$: $[(3)(-2)][(1)(3)(-2)] = (-6)(-6) = 36$.
Evaluate $\frac{y(2ab)}{-yb}$ for $a=2, b=4, y=-1$: $\frac{(-1)(2)(2)(4)}{-(-1)(4)} = \frac{-16}{4} = -4$.

Explanation

Think of variables as empty boxes labeled 'x' or 'y'. Evaluating is just plugging the given numbers into the correct boxes. Remember to use parentheses to keep negative and subtraction signs from getting mixed up, then calculate the result!

Property

An expression can be simplified before it is evaluated. Use tools like the Distributive Property to combine terms first.

Examples

Simplify $-a(b-2)+b$ for $a=0.5, b=-2.5$: First, $-ab+2a+b$. Then, $-(0.5)(-2.5)+2(0.5)+(-2.5) = 1.25+1-2.5 = -0.25$.
Simplify $y(y+2z)-y$ for $y=\frac{1}{3}, z=\frac{1}{6}$: First, $y^2+2yz-y$. Then, $(\frac{1}{3})^2+2(\frac{1}{3})(\frac{1}{6})-\frac{1}{3} = \frac{1}{9}+\frac{1}{9}-\frac{1}{3} = -\frac{1}{9}$.

Explanation

Why wrestle with a messy octopus of an expression? Tidy it up by simplifying first! It makes the final calculation much quicker and you are far less likely to make a silly mistake with all those numbers.

Property

In an expression like $ym^3$, only the variable directly next to the exponent is raised to the power. Use parentheses, as in $(ym)^3$, to apply the power to the entire group.

Examples

Evaluate $ab^3$ for $a=3$ and $b=-2$: $3(-2)^3 = 3(-8) = -24$. Notice only the $-2$ is cubed.
Evaluate $2(5-\frac{a}{b})^2$ for $a=6, b=-3$: $2(5-\frac{6}{-3})^2 = 2(5-(-2))^2 = 2(7)^2 = 2(49) = 98$.
Evaluate $|(-b)^3|$ for $b=4$: $|(-4)^3| = |-64| = 64$. The exponent works before the absolute value.

Explanation

Exponents are picky! They only power up their immediate neighbor. To give a negative sign or a whole group a power-up, you must wrap them in parentheses, like giving them a team uniform before the big game.