2.2 Evaluate, Simplify, and Translate Expressions

Learning objectives

Learning objectives

• Evaluate algebraic expressions

• Identify terms, coefficients, and like terms

• Simplify expressions by combining like terms

• Translate word phrases to algebraic expressions

Evaluate Algebraic Expressions

In the last section, we simplified expressions using the order of operations. In this section, we’ll evaluate expressions—again following the order of operations.

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.

Example 2.13

Solution

1. Recall the substitution rule. To evaluate an expression, replace the variable with its given value and then simplify the arithmetic.
2. Evaluate part ⓐ. Substituting $x=3$ gives $3+7=10$, so the expression equals $10$.
3. Evaluate part ⓑ. Substituting $x=12$ gives $12+7=19$, so the expression equals $19$.
4. Compare the results. The same expression produced two values because the assigned value of $x$ differed between the two parts.

Takeaway: Evaluating an expression means substituting the given value for its variable, so $x+7$ equals $10$ when $x=3$ and $19$ when $x=12$.

Example 2.14

Solution

1. Recall the rule for evaluation. Replace every occurrence of the variable with the given number before performing any arithmetic.
2. Evaluate part ⓐ. Substituting $x=5$ gives $9\cdot{}5-2=45-2=43$.
3. Evaluate part ⓑ. Substituting $x=1$ gives $9(1)-2=9-2=7$.
4. Check the order of operations. In both parts the multiplication was completed before the subtraction, matching the structure of the original expression.

Takeaway: Substitute before multiplying in $9x-2$: when $x=5$ the value is $43$, and when $x=1$ the value is $7$.

Example 2.15

Solution

1. Identify the base and exponent. In $x^2$, the variable is the base and the exponent $2$ tells how many factors to multiply.
2. Substitute the value. Replacing $x$ with $10$ gives the power $10^2$.
3. Evaluate the power. Squaring $10$ means multiplying $10$ by itself, so $10^2=10\cdot{}10=100$.
4. State the value. The expression $x^2$ has the value $100$ when $x=10$.

Takeaway: To evaluate $x^2$ at $x=10$, square the substituted value, giving $10^2=100$.

Example 2.16

Solution

1. Identify where the variable sits. In $2^x$, the variable is the exponent, so its value must be substituted before the power is evaluated.
2. Substitute the value. Replacing $x$ with $5$ gives the power $2^5$.
3. Expand the power. Five factors of $2$ give $2\cdot{}2\cdot{}2\cdot{}2\cdot{}2=32$.
4. State the value. The expression $2^x$ has the value $32$ when $x=5$.

Takeaway: When the variable is the exponent, substitute its value first and then evaluate the resulting power, so $2^5=32$.

Example 2.17

Solution

1. Plan the substitutions. The expression contains two variables, so both given values must replace their variables before any arithmetic is done.
2. Substitute the values. Replacing $x$ with $10$ and $y$ with $2$ gives $3(10)+4(2)-6$.
3. Multiply each term. The products are $3(10)=30$ and $4(2)=8$, giving $30+8-6$.
4. Combine the terms. Adding first gives $38$, and subtracting $6$ leaves $32$.
5. State the value. The expression equals $32$ when $x=10$ and $y=2$.

Takeaway: An expression with two variables requires one substitution for each variable, and $3(10)+4(2)-6=32$ when $x=10$ and $y=2$.

Example 2.18

Solution

1. Substitute the value. Replace each $x$ with $4$, obtaining $2(4)^2+3(4)+8$.
2. Evaluate the exponent. Since $4^2=16$, the expression becomes $2(16)+3(4)+8$.
3. Multiply each term. Compute $2(16)=32$ and $3(4)=12$, giving $32+12+8$.
4. Add the numerical terms. The figures show $32+12+8$ and the final value $52$, so $2x^2+3x+8=52$ when $x=4$.

Takeaway: In $2x^2$, the exponent applies only to $x$, so evaluate $x^2$ before multiplying it by the coefficient $2$.

Identify Terms, Coefficients, and Like Terms

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Some examples of terms are $7,y,5{x}^{2},9a,\mathrm{and}\,13xy.$

The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number in front of the variable. The coefficient of the term $3x$ is $3.$ When we write $x,$ the coefficient is $1,$ since $x=1\cdot{}x.$ Table 2.5 gives the coefficients for each of the terms in the left column.

Table

2.5

An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. Table 2.6 gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.

Table

2.6

Example 2.19

Solution

1. Separate the expression into terms. The addition signs show four terms: $9b$, $15x^2$, $a$, and $6$.
2. Identify the coefficient of $9b$. The coefficient is the numerical factor multiplying the variable, so the coefficient of $9b$ is $9$.
3. Identify the coefficient of $15x^2$. The numerical factor in front of $x^2$ is $15$, so the coefficient is $15$.
4. Identify the coefficient of $a$. Since no number is written before $a$, its coefficient is the understood number $1$.
5. Identify the coefficient of $6$. A constant is its own coefficient, so the coefficient of $6$ is $6$.
6. Collect the results. The terms are $9b$, $15x^2$, $a$, and $6$, with coefficients $9$, $15$, $1$, and $6$.

Takeaway: The terms of $9b+15x^2+a+6$ are $9b$, $15x^2$, $a$, and $6$, with coefficients $9$, $15$, $1$, and $6$ respectively.

Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?

Which of these terms are like terms?

• The terms $7$ and $4$ are both constant terms.

• The terms $5x$ and $3x$ are both terms with $x.$

• The terms ${n}^{2}$ and $9{n}^{2}$ both have ${n}^{2}.$

Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms $5x,7,{n}^{2},4,3x,9{n}^{2},$

Example 2.20

Solution

1. Recall the definition of like terms. Two terms are like only when their variables and exponents match exactly, and constants are like with constants.
2. Group part ⓐ. The like groups are $y^3$ with $4y^3$, $7x^2$ with $5x^2$, and $14$ with $23$, while $9x$ has no partner.
3. Group part ⓑ. The like groups are $4x^2$ with $5x^2$, and $2x$, $6x$, $40x$ together, while $8xy$ has no partner.
4. Check each grouping. Every paired group has an identical variable part, and each unpaired term differs in its variable or its exponent.

Takeaway: Like terms must have exactly the same variable part and exponent, so $y^3$, $x^2$, and constant terms form separate like-term groups, while $8xy$ has no partner.

Simplify Expressions by Combining Like Terms

We can simplify an expression by combining the like terms. What do you think $3x+6x$ would simplify to? If you thought $9x,$ you would be right!

We can see why this works by writing both terms as addition problems.

Add the coefficients and keep the same variable. It doesn’t matter what $x$ is. If you have $3$ of something and add $6$ more of the same thing, the result is $9$ of them. For example, $3$ oranges plus $6$ oranges is $9$ oranges. We will discuss the mathematical properties behind this later.

The expression $3x+6x$ has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms.

Now it is easier to see the like terms to be combined.

Example 2.21

Solution

1. Group the variable terms. The terms $3x$ and $4x$ have the same variable part, so they combine to $(3+4)x=7x$.
2. Group the constant terms. The terms $7$ and $5$ combine to $7+5=12$.
3. Write the simplified expression. The figures show $3x+7+4x+5$ regrouped as $3x+4x+7+5$ and then $7x+12$, so the simplest form is $7x+12$.

Takeaway: Only like terms can be combined, so rearrange an expression to place matching variable parts and matching constants together.

When any of the terms have negative coefficients, the procedure is the same, except that you have to subtract instead of adding to combine like terms.

Example 2.22

Solution

1. Group the squared terms. Combine $7x^2-x^2=(7-1)x^2=6x^2$.
2. Group the first-degree terms. Combine $8x-4x=(8-4)x=4x$.
3. Write the result. The figures show the regrouped expression $7x^2-x^2+8x-4x$ and the simplified form $6x^2+4x$. Since $6x^2$ and $4x$ are not like terms, the simplest form is $6x^2+4x$.

Takeaway: Terms with different exponents are not like terms, even when they contain the same variable.

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 2.7.

Table

2.7

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 2.23

Solution

1. Identify the key words. The word difference signals subtraction and the word quotient signals division.
2. Translate part ⓐ. The phrase the difference of $20$ and $4$ means $20$ subtracted by $4$, giving the expression $20-4$.
3. Translate part ⓑ. The phrase the quotient of $10x$ and $3$ means $10x$ divided by $3$, giving the expression $\frac{10x}{3}$.
4. Check the order. In each expression the first named quantity comes first and the second named quantity provides the second operand.

Takeaway: The key words difference and quotient indicate subtraction and division, giving $20-4$ and $\frac{10x}{3}$ respectively.

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 2.24

Solution

1. Identify the key phrases. More than means the stated amount is added to the quantity that follows, and less than means the stated amount is subtracted from the quantity that follows.
2. Translate part ⓐ. Eight more than $y$ means eight added to $y$, giving the expression $y+8$.
3. Translate part ⓑ. Seven less than $9z$ means seven subtracted from $9z$, giving the expression $9z-7$.
4. Check the order. Both expressions keep the original quantity in front and place the stated amount on the correct side of the operation.

Takeaway: In phrases using more than or less than, the stated quantity is added to or subtracted from the following expression, so the results are $y+8$ and $9z-7$.

Example 2.25

Solution

1. Identify the two operations. The words times and sum indicate multiplication and addition, so their order in the phrase decides the order of operations.
2. Translate part ⓐ. Five times the sum of $m$ and $n$ multiplies the whole sum, so the expression is $5(m+n)$.
3. Translate part ⓑ. The sum of five times $m$ and $n$ adds two separate terms, so the expression is $5m+n$.
4. Compare the expressions. Part ⓐ adds before multiplying while part ⓑ multiplies before adding, so the parentheses change the value.

Takeaway: Parentheses preserve the order stated in a word phrase: $5(m+n)$ adds $m$ and $n$ before multiplying, while $5m+n$ multiplies $m$ by $5$ before adding $n$.

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 2.26

Solution

1. Identify the variable quantity. Since $w$ represents the width, the height must be expressed in terms of $w$.
2. Rewrite the phrase. The phrase six inches less than the width means six inches subtracted from the width.
3. Substitute the variable. Replacing the word width with $w$ gives the phrase six inches subtracted from $w$.
4. Write the expression. The height is therefore represented by $w-6$, and the order of the subtraction follows the wording.

Takeaway: The phrase “$6$ less than $w$” means that $6$ is subtracted from $w$, so the height is represented by $w-6$.

Example 2.27

Solution

1. Identify the variable quantity. Since $q$ represents the number of quarters, the number of dimes must be written in terms of $q$.
2. Translate the product. Five times the number of quarters becomes $5q$ when $q$ replaces the phrase number of quarters.
3. Translate the subtraction. The phrase two less than applies to that whole product, so two is subtracted from $5q$.
4. Write the expression. The number of dimes is therefore represented by $5q-2$, which multiplies first and then subtracts.

Takeaway: When a quantity is $2$ less than $5$ times another quantity, multiply first and then subtract $2$, giving $5q-2$ dimes.

Section 2.2 Exercises

Section 2.2 Exercises

Evaluate Algebraic Expressions

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

Identify Terms, Coefficients, and Like Terms

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

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

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

Simplify Expressions by Combining Like Terms

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

Translate English Phrases into Algebraic Expressions

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

In the following exercises, write an algebraic expression.

In the following exercises, use algebraic expressions to solve the problem.

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

1. Evaluate:

$y+4\,\mathrm{when}\,$

1. ⓐ $\,y=6\,\,$

2. ⓑ $\,y=15$

2. Evaluate:

$a-5\,\mathrm{when}\,$

1. ⓐ $\,a=9\,\,$

2. ⓑ $\,a=17$

3. Evaluate:

$8x-3,\mathrm{when}\,$

1. ⓐ $\,x=2\,\,$

2. ⓑ $\,x=1$

1. Evaluate:

$4y-4,\mathrm{when}\,$

1. ⓐ $\,y=3\,\,$

2. ⓑ $\,y=5$

2. Identify all terms in the given expression, and their coefficients:

$4x+3b+2$

3. Identify all terms in the given expression, and their coefficients:

$9a+13{a}^{2}+{a}^{3}$

1. Identify the like terms in the list or the expression:

$9,2{x}^{3},{y}^{2},8{x}^{3},15,9y,11{y}^{2}$

2. Identify the like terms in the list or the expression:

$4{x}^{3}+8{x}^{2}+19+3{x}^{2}+24+6{x}^{3}$

3. Simplify:

$7x+9+9x+8$

1. Simplify:

$5y+2+8y+4y+5$

2. Simplify:

$3{x}^{2}+9x+{x}^{2}+5x$

3. Simplify:

$11{y}^{2}+8y+{y}^{2}+7y$

1. Translate the given word phrase into an algebraic expression:

1. ⓐ the difference of $47$ and $41$

2. ⓑ the quotient of $5x$ and $2$

2. Translate the given word phrase into an algebraic expression:

1. ⓐ the sum of $17$ and $19$

2. ⓑ the product of $7$ and $x$

3. Translate each word phrase into an algebraic expression:

1. ⓐ Eleven more than $x$

2. ⓑ Fourteen less than $11a$

1. Translate each word phrase into an algebraic expression:

1. ⓐ $19$ more than $j$

2. ⓑ $21$ less than $2x$

2. 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 $q$

3. Translate the word phrase into an algebraic expression:

1. ⓐ the difference of two times $x\,\mathrm{and 8}\,$

2. ⓑ two times the difference of $x\,\mathrm{and}\,8$

1. 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.

2. 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.

3. 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.

1. 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.

2. $15{x}^{2}+6x+2$

3. $10{y}^{3}+y+2$

1. ${x}^{3},8x,14,8y,5,8{x}^{3}$

2. $9a,{a}^{2},16ab,16{b}^{2},4ab,9{b}^{2}$

3. $10x+3x$

1. $17a+9a$

2. $4c+2c+c$

3. $9x+3x+8$

1. $7u+2+3u+1$

2. $7p+6+5p+4$

3. $10a+7+5a-2+7a-4$

1. $3{x}^{2}+12x+11+14{x}^{2}+8x+5$

2. The sum of 8 and 12

3. The difference of 14 and 9

1. The product of 9 and 7

2. The quotient of 36 and 9

3. The difference of $x$ and $4$

1. The product of $6$ and $y$

2. The sum of $8x$ and $3x$

3. The quotient of $y$ and $3$

1. Eight times the difference of $y$ and nine

2. Five times the sum of $x$ and $y$

3. Adele bought a skirt and a blouse. The skirt cost $\text{\textdollar 15}$ more than the blouse. Let $b$ represent the cost of the blouse. Write an expression for the cost of the skirt.

1. The number of girls in a second-grade class is $4$ less than the number of boys. Let $b$ represent the number of boys. Write an expression for the number of girls.

2. Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let $n$ represent the number of nickels. Write an expression for the number of pennies.

3. Car insurance Justin’s car insurance has a $\text{\textdollar 750}$ deductible per incident. This means that he pays $\text{\textdollar 750}$ and his insurance company will pay all costs beyond $\text{\textdollar 750.}$ If Justin files a claim for $\text{\textdollar 2,100,}$ how much will he pay, and how much will his insurance company pay?