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Expressions with One or More Variables

Evaluate expressions given values for variables.

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Practice Expressions with One or More Variables
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Expressions with One or More Variables

Credit: maaco
Source: https://www.flickr.com/photos/mako_side_b/2409621948

Lisa's older sister Theresa has agreed to loan Lisa some money so Lisa can buy a new pair of jeans. Theresa is going to charge Lisa interest until Lisa pays her back. Theresa tells Lisa that she will calculate how much interest Lisa owes after t\begin{align*}t\end{align*} months according to the expression 110jt\begin{align*}\frac{1}{10} \cdot j \cdot t\end{align*} where j\begin{align*}j\end{align*} is the cost of the jeans. Lisa wants to figure out how much interest she will owe Theresa if the jeans cost 60 and she doesn't pay Theresa back for 3 months. In this concept, you will learn how to evaluate expressions with more than one variable. Guidance A variable is a symbol or letter (such as x,m,R,y,P,\begin{align*}x,m,R,y,P,\end{align*} or a\begin{align*}a\end{align*}) that is used to represent a quantity that might change in value. A variable expression is a mathematical phrase that contains numbers, operations, and variables. Here are some examples of variable expressions: • 3x+y\begin{align*}3x+y\end{align*} • 10rx\begin{align*}10r-x\end{align*} • b3+2\begin{align*}b^3+2\end{align*} • mx4\begin{align*}mx-4\end{align*} You can use a variable expression to describe a real world situation where one or more quantities has an unknown value or can change in value. To evaluate a variable expression means to find the value of the expression for given values of the variables. To evaluate, substitute the given values for the variables in the expression and simplify using the order of operations. To follow the order of operations, you always need to do any multiplication/division first before any addition/subtraction. Here is an example. Evaluate the expression xy+x\begin{align*}xy+x\end{align*} if x=2\begin{align*}x=2\end{align*} and y=4\begin{align*}y=4\end{align*}. First, substitute 2 in for the letter x\begin{align*}x\end{align*} and 4 in for the letter y\begin{align*}y\end{align*} in the expression. (2)(4)+2 Notice that you can put parentheses around the numbers to keep them separate and to indicate multiplication. Now, simplify the expression using the order of operations. You will need to multiply first and then add. (2)(4)+2==8+210 The answer is 10. Here is another example that involves fractions. Evaluate the expression mx+3m\begin{align*}mx+3m\end{align*} for x=23\begin{align*}x=\frac{2}{3}\end{align*} and m=9\begin{align*}m=9\end{align*}. First, substitute 23\begin{align*}\frac{2}{3}\end{align*} in for the letter x\begin{align*}x\end{align*} and 9 in for the letter m\begin{align*}m\end{align*} in the expression. (9)(23)+3(9) Now, simplify the expression using the order of operations. You will need to multiply each part of the expression first. Remember, to multiply a whole number times a fraction, you can first turn the whole number into a fraction by writing it over 1. Then, multiply the numerators and multiply the denominators. Finally, simplify the fraction. (9)(23)+3(9)===9123+27183+276+27 Next, continue to simplify the expression using the order of operations by adding the two terms. 6+27=33 The answer is 33. Guided Practice Evaluate the expression xy+xy\begin{align*}xy+xy\end{align*} for x=12\begin{align*}x=\frac{1}{2}\end{align*} and y=23\begin{align*}y=\frac{2}{3}\end{align*}. First, substitute 12\begin{align*}\frac{1}{2}\end{align*} in for the letter x\begin{align*}x\end{align*} and 23\begin{align*}\frac{2}{3}\end{align*} in for the letter y\begin{align*}y\end{align*} in the expression. (12)(23)+(12)(23) Now, simplify the expression using the order of operations. You will need to multiply each part of the expression first. Remember, to multiply fractions you should multiply the numerators and multiply the denominators. Then, simplify the fraction. Next, continue to simplify the expression using the order of operations by adding the two terms. Remember, to add fractions you need a common denominator. Here, you already have a common denominator of 3. So, add the numerators of your fractions and keep the common denominator as your denominator. The answer is \begin{align*}\frac{2}{3}\end{align*}. Examples Example 1 Evaluate the expression \begin{align*}ab+7\end{align*} when \begin{align*}a\end{align*} is 9 and \begin{align*}b\end{align*} is 8. First, substitute 9 in for the letter \begin{align*}a\end{align*} and 8 in for the letter \begin{align*}b\end{align*}. Now, simplify the expression using the order of operations. You will need to multiply first and then add. The answer is 79. Example 2 Evaluate the expression \begin{align*}xy+zx\end{align*} when \begin{align*}x\end{align*} is 2, \begin{align*}y\end{align*} is 5, and \begin{align*}z\end{align*} is 7. First, substitute 2 in for the letter \begin{align*}x\end{align*}, 5 in for the letter \begin{align*}y\end{align*}, and 7 in for the letter \begin{align*}z\end{align*}. Now, simplify the expression using the order of operations. You will need to multiply each part of the expression first and then add. The answer is 24. Example 3 Evaluate the expression \begin{align*}xy+x\end{align*} when \begin{align*}x\end{align*} is \begin{align*}\frac{1}{4}\end{align*} and \begin{align*}y\end{align*} is 3. First, substitute \begin{align*}\frac{1}{4}\end{align*} in for the letter \begin{align*}x\end{align*} and 3 in for the letter \begin{align*}y\end{align*} in the expression. Now, simplify the expression using the order of operations. You will need to multiply the first part of the expression first. Remember, to multiply a whole number and a fraction, you can first turn the whole number into a fraction by writing it over 1. Then, multiply the numerators and multiply the denominators. Next, continue to simplify the expression using the order of operations by adding the two terms. Remember, to add fractions you need a common denominator. Here, you already have a common denominator of 4. So, add the numerators of your fractions and keep the common denominator as your denominator. Then, simplify. The answer is 1. Follow Up Credit: stopnlook Source: https://www.flickr.com/photos/crazyneighborlady/415534585 License: CC BY-NC 3.0 Remember Lisa and the new pair of jeans she wants to buy? Her sister Theresa will loan her the money to buy them, but she will need to pay her sister back plus interest. Her sister will use the following expression to calculate how much interest Lisa owes her: In this expression, \begin{align*}j\end{align*} represents the cost of the jeans and \begin{align*}t\end{align*} represents the number of months before Lisa pays Theresa back. Lisa wants to figure out how much interest she will owe Theresa if the jeans cost60 and she pays Theresa back in 3 months.

First, Lisa should substitute 60 in for the letter \begin{align*}j\end{align*} and 3 in for the letter \begin{align*}t\end{align*} in the expression.

Now, simplify the expression using the order of operations. You will need to multiply \begin{align*}\frac{1}{10}\end{align*} by 60 and then multiply that result by 3.

Remember, to multiply a whole number and a fraction, you can first turn the whole number into a fraction by writing it over 1. Then, multiply the numerators and multiply the denominators. Finally, simplify the fraction.

The answer is that Lisa will owe Theresa \$18 in interest if she waits 3 months to pay Theresa back. This means in 3 months she will owe her sister a total of \begin{align*}60+18=\78\end{align*} for the jeans and the interest!

Explore More

Evaluate each expression if \begin{align*}x = 3\end{align*} and \begin{align*}y = 4\end{align*}.

1. \begin{align*}xy + 2y\end{align*}

2. \begin{align*}3y + 2y\end{align*}

3. \begin{align*}3y + 9x\end{align*}

4. \begin{align*}xy + 3xy\end{align*}

5. \begin{align*}2xy + 9xy\end{align*}

Evaluate each expression if \begin{align*}x = 10\end{align*} and \begin{align*}y = 5\end{align*}.

6. \begin{align*}5x + xy\end{align*}

7. \begin{align*}3x + 2y\end{align*}

8. \begin{align*}2x + 3y\end{align*}

9. \begin{align*}4x + 3y + x\end{align*}

10. \begin{align*}5x + 3y + 2x\end{align*}

Evaluate each expression if \begin{align*}y = 2\end{align*} and \begin{align*}z = 4\end{align*}.

11. \begin{align*}2y + 3y- 2z\end{align*}

12. \begin{align*}5y + 3y- 3z\end{align*}

13. \begin{align*}2y + 5y- 2z\end{align*}

14. \begin{align*}4y + 3y- 2z\end{align*}

15. Evaluate \begin{align*}6z- 2(z + x)\end{align*} if \begin{align*}x\end{align*} is 3 and \begin{align*}z\end{align*} is 4.

16. Evaluate \begin{align*}8a + 3b- 2c\end{align*} if \begin{align*}a\end{align*} is 5, \begin{align*}b\end{align*} is 4 and \begin{align*}c\end{align*} is 3.

Vocabulary Language: English

algebraic

algebraic

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

Algebraic Expression

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

Exponent

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

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

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

substitute

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

Variable

A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.