# 1.2: Expressions with One or More Variables

**At Grade**Created by: CK-12

**Practice**Expressions with One or More Variables

Suppose you had a summer job that paid you $25 per week, plus $10 per hour. Your weekly paycheck could be represented as \begin{align*}10h+25,\end{align*}

### Watch This

CK-12 Foundation: 0102s evaluate algebraic expressions

### Guidance

When given an algebraic expression, one of the most common things to do with it is **evaluate** it for some given value of the variable.

Take a look at this example to see how this works:

#### Example A

*Let x = 12. Find the value of 2x - 7.*

**Solution:**

To find the solution, substitute 12 in place of \begin{align*}x\end{align*}

\begin{align}
2x - 7 & = 2(12) - 7\\
& = 24 - 7\\
& = 17
\end{align}

**Note:** In the first step of the problem, keep the substituted value in parentheses. This makes the written-out problem easier to follow, and helps avoid mistakes. (If we didn’t use parentheses and also forgot to add a multiplication sign, we would end up turning "\begin{align*}2x\end{align*}*times* 12!")

#### Example B

*Let \begin{align*}y = -2. \end{align*} y=−2. Find the value of \begin{align*} \frac {7} {y} - 11 y + 2.\end{align*}7y−11y+2.*

**Solution**

\begin{align}
\frac {7} {(-2)} - 11( -2 ) + 2 & = -3 \frac { 1 } { 2 } + 22 + 2\\
& = 24 - 3 \frac { 1 } { 2 }\\
& = 20 \frac { 1 } { 2 }
\end{align}

Many expressions have more than one variable in them. For example, the formula for the perimeter of a rectangle, \begin{align*}P=2l+2w\end{align*}

#### Example C

*The area of a trapezoid is given by the equation \begin{align*} A = \frac{ h } { 2 } (a + b)\end{align*} A=h2(a+b). Find the area of a trapezoid with bases \begin{align*}a = 10 \ cm\end{align*}a=10 cm and \begin{align*}b = 15 \ cm\end{align*}b=15 cm and height \begin{align*}h = 8 \ cm\end{align*}h=8 cm.*

**Solution:**

To find the solution to this problem, substitute the given values for variables \begin{align*}a, \ b,\end{align*}

\begin{align}
A & = \frac{h}{2}(a+b) \qquad \qquad \quad \text{ Substitute } 10 \text{ for } a, \ 15 \text{ for }b, \ \text{and } 8 \text{ for } h. \\
& = \frac{(8)}{2} \left( (10)+(15) \right) \qquad \text{Evaluate piece by piece. } 10+15=25 \text{ and } \frac{8}{2}=4. \\
& = 4(25) \\
A & = 100 \\
\end{align}

You may wish to watch this video for help with the examples above.

CK-12 Foundation: Evaluate Algebraic Expressions

### Guided Practice

*Let \begin{align*}x= 3\end{align*} x=3 and \begin{align*}y = -2. \end{align*}y=−2. Find the value of \begin{align*} 3xy + \frac{6}{y}-2x \end{align*}3xy+6y−2x.*

**Solution**

\begin{align}
3xy + \frac{6}{y}-2x & = 3(3)(-2) + \frac{6}{(-2)}-2(3)\\
& = (-18)+(-3)-(6)\\
& = -27
\end{align}

### Explore More

Evaluate questions 1 through 8, using \begin{align*}a = -3, \ b = 2, \ c = 5,\end{align*}

- \begin{align*}2a + 3b\end{align*}
2a+3b - \begin{align*}4c + d\end{align*}
4c+d - \begin{align*}5ac - 2b\end{align*}
5ac−2b - \begin{align*} \frac { 2a } { c - d }\end{align*}
2ac−d - \begin{align*} \frac { 3b } { d }\end{align*}
3bd - \begin{align*} \frac { a - 4b } { 3c + 2d }\end{align*}
a−4b3c+2d - \begin{align*} \frac { 1 } { a + b }\end{align*}
1a+b - \begin{align*} \frac { ab } {cd }\end{align*}
abcd

For questions 9 through 11, the weekly cost \begin{align*}C\end{align*}

- What is the cost of producing 1000 remote controls?
- What is the cost of producing 2000 remote controls?
- What is the cost of producing 2500 remote controls?

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 1.2.

algebraic

The word*algebraic*indicates that a given expression or equation includes variables.

Algebraic Expression

An expression that has numbers, operations and variables, but no equals sign.Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.Expression

An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.Order of Operations

The order of operations specifies the order in which to perform each of multiple operations in an expression or equation. The order of operations is: P - parentheses, E - exponents, M/D - multiplication and division in order from left to right, A/S - addition and subtraction in order from left to right.Parentheses

Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.substitute

In algebra, to substitute means to replace a variable or term with a specific value.Trapezoid

A trapezoid is a quadrilateral with exactly one pair of parallel opposite sides.### Image Attributions

Here you'll learn how to evaluate algebraic expressions by plugging in specific values for its variable(s).

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