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2.2: Evaluate, Simplify, and Translate Expressions (Part 2)

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• Mathematics at OpenStax CNX

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Translate Words to Algebraic Expressions

In the previous section, we listed many operation symbols that are used in algebra, and then we translated expressions and equations into word phrases and sentences. Now we’ll reverse the process and translate word phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. They are summarized in Table $$\PageIndex{3}$$.

Table $$\PageIndex{3}$$
Operation Phrase Expression

a plus b

the sum of a and b

a increased by b

b more than a

the total of a and b

a + b
Subtraction

a minus b

the difference of a and b

b subtracted from a

a decreased by b

b less than a

a - b
Multiplication

a times b

the product of a and b

a • b, ab, a(b), (a)(b)
Division

a divided by b

the quotient of a and b

the ratio of a and b

b divided into a

a ÷ b, a / b, $$\frac{a}{b}$$, $$b \overline{) a}$$

Look closely at these phrases using the four operations:

• the sum of $$a$$ and $$b$$
• the difference of $$a$$ and $$b$$
• the product of $$a$$ and $$b$$
• the quotient of $$a$$ and $$b$$

Each phrase tells you to operate on two numbers. Look for the words of and and to find the numbers.

Example $$\PageIndex{11}$$: translate

Translate each word phrase into an algebraic expression:

1. the difference of $$20$$ and $$4$$
2. the quotient of $$10x$$ and $$3$$

Solution

1. The key word is difference, which tells us the operation is subtraction. Look for the words of and and to find the numbers to subtract.

the difference of $$20$$ and $$4$$

$$20$$ minus $$4$$

$$20 − 4$$

1. The key word is quotient, which tells us the operation is division.

the quotient of $$10x$$ and $$3$$

divide $$10x$$ by $$3$$

$$10x ÷ 3$$

This can also be written as 1$$0x / 3$$ or $$\frac{10x}{3}$$

exercise $$\PageIndex{21}$$

Translate the given word phrase into an algebraic expression:

1. the difference of $$47$$ and $$41$$
2. the quotient of $$5x$$ and $$2$$

$$47-41$$

$$5x\div 2$$

exercise $$\PageIndex{22}$$

Translate the given word phrase into an algebraic expression:

1. the sum of $$17$$ and $$19$$
2. the product of $$7$$ and $$x$$

$$17+19$$

$$7x$$

How old will you be in eight years? What age is eight more years than your age now? Did you add $$8$$ to your present age? Eight more than means eight added to your present age.

How old were you seven years ago? This is seven years less than your age now. You subtract $$7$$ from your present age. Seven less than means seven subtracted from your present age.

Example $$\PageIndex{12}$$: translate

Translate each word phrase into an algebraic expression:

1. Eight more than $$y$$
2. Seven less than $$9z$$

Solution

1. The key words are more than. They tell us the operation is addition. More than means “added to”.

Eight more than $$y$$

Eight added to $$y$$

$$y + 8$$

1. The key words are less than. They tell us the operation is subtraction. Less than means “subtracted from”.

Seven less than $$9z$$

Seven subtracted from $$9z$$

$$9z − 7$$

exercise $$\PageIndex{23}$$

Translate each word phrase into an algebraic expression:

1. Eleven more than $$x$$
2. Fourteen less than $$11a$$

$$x+11$$

$$11a-14$$

exercise $$\PageIndex{24}$$

Translate each word phrase into an algebraic expression:

1. $$19$$ more than $$j$$
2. $$21$$ less than $$2x$$

$$j+19$$

$$2x-21$$

Example $$\PageIndex{13}$$: translate

Translate each word phrase into an algebraic expression:

1. five times the sum of $$m$$ and $$n$$
2. the sum of five times $$m$$ and $$n$$

Solution

1. There are two operation words: times tells us to multiply and sum tells us to add. Because we are multiplying $$5$$ times the sum, we need parentheses around the sum of $$m$$ and $$n$$.

five times the sum of $$m$$ and $$n$$

$$5(m + n)$$

1. To take a sum, we look for the words of and and to see what is being added. Here we are taking the sum of five times $$m$$ and $$n$$.

the sum of five times $$m$$ and $$n$$

$$5m + n$$

Notice how the use of parentheses changes the result. In part (a), we add first and in part (b), we multiply first.

exercise $$\PageIndex{25}$$

Translate the word phrase into an algebraic expression:

1. four times the sum of $$p$$ and $$q$$
2. the sum of four times $$p$$ and $$q$$

$$4(p+q)$$

$$4p+q$$

exercise $$\PageIndex{26}$$

Translate the word phrase into an algebraic expression:

1. the difference of two times $$x$$ and $$8$$
2. two times the difference of $$x$$ and $$8$$

$$2x-8$$

$$2(x-8)$$

Later in this course, we’ll apply our skills in algebra to solving equations. We’ll usually start by translating a word phrase to an algebraic expression. We’ll need to be clear about what the expression will represent. We’ll see how to do this in the next two examples.

Example $$\PageIndex{14}$$: write an expression

The height of a rectangular window is 6 inches less than the width. Let w represent the width of the window. Write an expression for the height of the window.

Solution

 Write a phrase about the height. 6 less than the width Substitute w for the width. 6 less than w Rewrite 'less than' as 'subtracted from'. 6 subtracted from w Translate the phrase into algebra. w - 6

exercise $$\PageIndex{27}$$

The length of a rectangle is $$5$$ inches less than the width. Let $$w$$ represent the width of the rectangle. Write an expression for the length of the rectangle.

$$w-5$$

exercise $$\PageIndex{28}$$

The width of a rectangle is $$2$$ meters greater than the length. Let $$l$$ represent the length of the rectangle. Write an expression for the width of the rectangle.

$$l+2$$

Example $$\PageIndex{15}$$: write an expression

Blanca has dimes and quarters in her purse. The number of dimes is $$2$$ less than $$5$$ times the number of quarters. Let $$q$$ represent the number of quarters. Write an expression for the number of dimes.

Solution

 Write a phrase about the number of dimes. two less than five times the number of quarters Substitute q for the number of quarters. 2 less than five times q Translate 5 times q. 2 less than 5q Translate the phrase into algebra. 5q - 2

exercise $$\PageIndex{29}$$

Geoffrey has dimes and quarters in his pocket. The number of dimes is seven less than six times the number of quarters. Let $$q$$ represent the number of quarters. Write an expression for the number of dimes.

$$6q-7$$

exercise $$\PageIndex{30}$$

Lauren has dimes and nickels in her purse. The number of dimes is eight more than four times the number of nickels. Let $$n$$ represent the number of nickels. Write an expression for the number of dimes.

$$4n+8$$

Key Concepts

• Combine like terms.
• Identify like terms.
• Rearrange the expression so like terms are together.
• Add the coefficients of the like terms

Glossary

term

A term is a constant or the product of a constant and one or more variables.

coefficient

The constant that multiplies the variable(s) in a term is called the coefficient.

like terms

Terms that are either constants or have the same variables with the same exponents are like terms.

evaluate

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number.

Practice Makes Perfect

Evaluate Algebraic Expressions

In the following exercises, evaluate the expression for the given value.

1. 7x + 8 when x = 2
2. 9x + 7 when x = 3
3. 5x − 4 when x = 6
4. 8x − 6 when x = 7
5. x2 when x = 12
6. x3 when x = 5
7. x5 when x = 2
8. x4 when x = 3
9. 3x when x = 3
10. 4x when x = 2
11. x2 + 3x − 7 when x = 4
12. x2 + 5x − 8 when x = 6
13. 2x + 4y − 5 when x = 7, y = 8
14. 6x + 3y − 9 when x = 6, y = 9
15. (x − y)2 when x = 10, y = 7
16. (x + y)2 when x = 6, y = 9
17. a2 + b2 when a = 3, b = 8
18. r2 − s2 when r = 12, s = 5
19. 2l + 2w when l = 15, w = 12
20. 2l + 2w when l = 18, w = 14

Identify Terms, Coefficients, and Like Terms

In the following exercises, list the terms in the given expression.

1. 15x2 + 6x + 2
2. 11x2 + 8x + 5
3. 10y3 + y + 2
4. 9y3 + y + 5

In the following exercises, identify the coefficient of the given term.

1. 8a
2. 13m
3. 5r2
4. 6x3

In the following exercises, identify all sets of like terms.

1. x3, 8x, 14, 8y, 5, 8x3
2. 6z, 3w2, 1, 6z2, 4z, w2
3. 9a, a2, 16ab, 16b2, 4ab, 9b2
4. 3, 25r2, 10s, 10r, 4r2, 3s

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the given expression by combining like terms.

1. 10x + 3x
2. 15x + 4x
3. 17a + 9a
4. 18z + 9z
5. 4c + 2c + c
6. 6y + 4y + y
7. 9x + 3x + 8
8. 8a + 5a + 9
9. 7u + 2 + 3u + 1
10. 8d + 6 + 2d + 5
11. 7p + 6 + 5p + 4
12. 8x + 7 + 4x − 5
13. 10a + 7 + 5a − 2 + 7a − 4
14. 7c + 4 + 6c − 3 + 9c − 1
15. 3x2 + 12x + 11 + 14x2 + 8x + 5
16. 5b2 + 9b + 10 + 2b2 + 3b − 4

Translate English Phrases into Algebraic Expressions

In the following exercises, translate the given word phrase into an algebraic expression.

1. The sum of 8 and 12
2. The sum of 9 and 1
3. The difference of 14 and 9
4. 8 less than 19
5. The product of 9 and 7
6. The product of 8 and 7
7. The quotient of 36 and 9
8. The quotient of 42 and 7
9. The difference of x and 4
10. 3 less than x
11. The product of 6 and y
12. The product of 9 and y
13. The sum of 8x and 3x
14. The sum of 13x and 3x
15. The quotient of y and 3
16. The quotient of y and 8
17. Eight times the difference of y and nine
18. Seven times the difference of y and one
19. Five times the sum of x and y
20. Nine times five less than twice x

In the following exercises, write an algebraic expression.

Writing Exercises

1. Explain why “the sum of x and y” is the same as “the sum of y and x,” but “the difference of x and y” is not the same as “the difference of y and x.” Try substituting two random numbers for x and y to help you explain. 146. Explain the difference between “4 times the sum of x and y” and “the sum of 4 times x and y.”

Self Check

(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. (b) After reviewing this checklist, what will you do to become confident for all objectives?