m81270 Evaluate, Simplify, and Translate Expressions By the end of this section, you will be able to: Evaluate algebraic expressions Identify terms, coefficients, and like terms Simplify expressions by combining like terms Translate word phrases to algebraic expressions f8ffe655-ee35-4be9-9b1b-ae94f43c84fa Before you get started, take this readiness quiz. Is n÷5 an expression or an equation?If you missed this problem, review . expression Simplify 45.If you missed this problem, review . 1,024 Simplify 1+8⋅9.If you missed this problem, review . 73

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. Evaluate x+7 when ⓐ x=3 ⓑ x=12 ⓐ To evaluate, substitute 3 for x in the expression, and then simplify. Substitute. Add. When x=3, the expression x+7 has a value of 10. ⓑ To evaluate, substitute 12 for x in the expression, and then simplify. Substitute. Add. When x=12, the expression x+7 has a value of 19. Notice that we got different results for parts ⓐ and ⓑ even though we started with the same expression. This is because the values used for x were different. When we evaluate an expression, the value varies depending on the value used for the variable. Evaluate: y+4when ⓐ y=6 ⓑ y=15 ⓐ 10 ⓑ 19 Evaluate: a−5when ⓐ a=9 ⓑ a=17 ⓐ 4 ⓑ 12 Evaluate 9x−2,when ⓐ x=5 ⓑ x=1 Remember ab means a times b, so 9x means 9 times x. ⓐ To evaluate the expression when x=5, we substitute 5 for x, and then simplify. Multiply. Subtract. ⓑ To evaluate the expression when x=1, we substitute 1 for x, and then simplify. Multiply. Subtract. Notice that in part ⓐ that we wrote 9⋅5 and in part ⓑ we wrote 9(1). Both the dot and the parentheses tell us to multiply. Evaluate: 8x−3,when ⓐ x=2 ⓑ x=1 ⓐ 13 ⓑ 5 Evaluate: 4y−4,when ⓐ y=3 ⓑ y=5 ⓐ 8 ⓑ 16 Evaluate x2 when x=10. We substitute 10 for x, and then simplify the expression. Use the definition of exponent. Multiply. When x=10, the expression x2 has a value of 100. Evaluate: x2whenx=8. 64 Evaluate: x3whenx=6. 216 Evaluate2xwhenx=5. In this expression, the variable is an exponent. Use the definition of exponent. Multiply. When x=5, the expression 2x has a value of 32. Evaluate: 2xwhenx=6. 64 Evaluate: 3xwhenx=4. 81 Evaluate3x+4y−6whenx=10andy=2. This expression contains two variables, so we must make two substitutions. Multiply. Add and subtract left to right. When x=10 and y=2, the expression 3x+4y−6 has a value of 32. Evaluate: 2x+5y−4whenx=11andy=3 33 Evaluate: 5x−2y−9whenx=7andy=8 10 Evaluate2x2+3x+8whenx=4. We need to be careful when an expression has a variable with an exponent. In this expression, 2x2 means 2⋅x⋅x and is different from the expression (2x)2, which means 2x⋅2x. Simplify 42. Multiply. Add. Evaluate: 3x2+4x+1whenx=3. 40 Evaluate: 6x2−4x−7whenx=2. 9

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,5x2,9a,and13xy. 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⋅x. gives the coefficients for each of the terms in the left column. Term Coefficient 9a 9 y 1 5x2 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. gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it. Expression Terms 7 7 y y x+7 x,7 2x+7y+4 2x,7y,4 3x2+4x2+5y+3 3x2,4x2,5y,3 Identify each term in the expression 9b+15x2+a+6. Then identify the coefficient of each term. The expression has four terms. They are 9b,15x2,a, and 6. The coefficient of 9b is 9. The coefficient of 15x2 is 15. Remember that if no number is written before a variable, the coefficient is 1. So the coefficient of a is 1. The coefficient of a constant is the constant, so the coefficient of 6 is 6. Identify all terms in the given expression, and their coefficients: 4x+3b+2 The terms are 4x, 3b, and 2. The coefficients are 4, 3, and 2. Identify all terms in the given expression, and their coefficients: 9a+13a2+a3 The terms are 9a, 13a², and a³, The coefficients are 9, 13, and 1. Some terms share common traits. Look at the following terms. Which ones seem to have traits in common? 5x,7,n2,4,3x,9n2 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 n2 and 9n2 both have n2. 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,n2,4,3x,9n2, 7and4are like terms. 5xand3xare like terms. n2and9n2are like terms. Terms that are either constants or have the same variables with the same exponents are like terms. Identify the like terms: ⓐ y3,7x2,14,23,4y3,9x,5x2 ⓑ 4x2+2x+5x2+6x+40x+8xy ⓐ y3,7x2,14,23,4y3,9x,5x2Look at the variables and exponents. The expression contains y3,x2,x, and constants. The terms y3 and 4y3 are like terms because they both have y3. The terms 7x2 and 5x2 are like terms because they both have x2. The terms 14 and 23 are like terms because they are both constants. The term 9x does not have any like terms in this list since no other terms have the variable x raised to the power of 1. ⓑ 4x2+2x+5x2+6x+40x+8xyLook at the variables and exponents. The expression contains the terms 4x2,2x,5x2,6x,40x,and8xy The terms 4x2 and 5x2 are like terms because they both have x2. The terms 2x,6x,and40x are like terms because they all have x. The term 8xy has no like terms in the given expression because no other terms contain the two variables xy. Identify the like terms in the list or the expression: 9,2x3,y2,8x3,15,9y,11y2 9 and 15; 2x³ and 8x³; y² and 11y² Identify the like terms in the list or the expression: 4x3+8x2+19+3x2+24+6x3 4x³ and 6x³; 8x² and 3x²; 19 and 24

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. Identify like terms. Rearrange the expression so like terms are together. Add the coefficients of the like terms. Simplify the expression: 3x+7+4x+5. Identify the like terms. Rearrange the expression, so the like terms are together. Add the coefficients of the like terms. The original expression is simplified to... Simplify: 7x+9+9x+8 16x + 17 Simplify: 5y+2+8y+4y+5 17y + 7 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. Simplify the expression: 7x2+8x–x2–4x. Identify the like terms. Rearrange the expression so like terms are together. Add the coefficients of the like terms. These are not like terms and cannot be combined. So 6x2+4x is in simplest form. Simplify: 3x2+9x+x2+5x 4x² + 14x Simplify: 11y2+8y+y2+7y 12y² + 15y

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 . Operation Phrase Expression Addition a plus b the sum of a and b a increased by b b more than a the total of a and b b added to a 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, ab, ba 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. Translate each word phrase into an algebraic expression: ⓐ the difference of 20 and 4 ⓑ the quotient of 10x and 3 ⓐ 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 differenceof20and420minus420−4 ⓑ The key word is quotient, which tells us the operation is division.the quotient of10xand3divide10xby310x÷3 This can also be written as 10x/3or10x3 Translate the given word phrase into an algebraic expression: ⓐ the difference of 47 and 41 ⓑ the quotient of 5x and 2 ⓐ 47 − 41 ⓑ 5x ÷ 2 Translate the given word phrase into an algebraic expression: ⓐ the sum of 17 and 19 ⓑ 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. Translate each word phrase into an algebraic expression: ⓐ Eight more than y ⓑ Seven less than 9z ⓐ The key words are more than. They tell us the operation is addition. More than means “added to”.Eight more thanyEight added toyy+8 ⓑ The key words are less than. They tell us the operation is subtraction. Less than means “subtracted from”.Seven less than9zSeven subtracted from9z9z−7 Translate each word phrase into an algebraic expression: ⓐ Eleven more than x ⓑ Fourteen less than 11a ⓐ x + 11 ⓑ 11a − 14 Translate each word phrase into an algebraic expression: ⓐ 19 more than j ⓑ 21 less than 2x ⓐ j + 19 ⓑ 2x − 21 Translate each word phrase into an algebraic expression: ⓐ five times the sum of m and n ⓑ the sum of five times m and n ⓐ 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)ⓑ 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+nNotice how the use of parentheses changes the result. In part ⓐ , we add first and in part ⓑ , we multiply first. Translate the word phrase into an algebraic expression: ⓐ four times the sum of p and q ⓑ the sum of four times p and q ⓐ 4(p + q) ⓐ 4p + q Translate the word phrase into an algebraic expression: ⓐ the difference of two times xand 8 ⓑ two times the difference of xand8 ⓐ 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. 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. 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 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 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 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. 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 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 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 Algebraic Expression Vocabulary

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

Evaluate Algebraic Expressions In the following exercises, evaluate the expression for the given value. 7x+8whenx=2 22 9x+7whenx=3 5x−4whenx=6 26 8x−6whenx=7 x2whenx=12 144 x3whenx=5 x5whenx=2 32 x4whenx=3 3xwhenx=3 27 4xwhenx=2 x2+3x−7whenx=4 21 x2+5x−8whenx=6 2x+4y−5whenx=7,y=8 41 6x+3y−9whenx=6,y=9 (x−y)2whenx=10,y=7 9 (x+y)2whenx=6,y=9 225 a2+b2whena=3,b=8 73 r2−s2whenr=12,s=5 2l+2wwhenl=15,w=12 54 2l+2wwhenl=18,w=14 Identify Terms, Coefficients, and Like Terms In the following exercises, list the terms in the given expression. 15x2+6x+2 15x², 6x, 2 11x2+8x+5 10y3+y+2 10y³, y, 2 9y3+y+5 In the following exercises, identify the coefficient of the given term. 8a 8 13m 5r2 5 6x3 In the following exercises, identify all sets of like terms. x3,8x,14,8y,5,8x3 x³ and 8x³; 14 and 5 6z,3w2,1,6z2,4z,w2 9a,a2,16ab,16b2,4ab,9b2 16ab and 4ab; 16b² and 9b² 3,25r2,10s,10r,4r2,3s Simplify Expressions by Combining Like Terms In the following exercises, simplify the given expression by combining like terms. 10x+3x 13x 15x+4x 17a+9a 26a 18z+9z 4c+2c+c 7c 6y+4y+y 9x+3x+8 12x + 8 8a+5a+9 7u+2+3u+1 10u + 3 8d+6+2d+5 7p+6+5p+4 12p + 10 8x+7+4x−5 10a+7+5a−2+7a−4 22a + 1 7c+4+6c−3+9c−1 3x2+12x+11+14x2+8x+5 17x² + 20x + 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. The sum of 8 and 12 8 + 12 The sum of 9 and 1 The difference of 14 and 9 14 − 9 8 less than 19 The product of 9 and 7 9 ⋅ 7 The product of 8 and 7 The quotient of 36 and 9 36 ÷ 9 The quotient of 42 and 7 The difference of x and 4 x − 4 3 less than x The product of 6 and y 6y The product of 9 and y The sum of 8x and 3x 8x + 3x The sum of 13x and 3x The quotient of y and 3 y ÷ 3 The quotient of y and 8 Eight times the difference of y and nine 8 (y − 9) Seven times the difference of y and one Five times the sum of x and y 5 (x + y) Nine times five less than twice x In the following exercises, write an algebraic expression. Adele bought a skirt and a blouse. The skirt cost $15 more than the blouse. Let b represent the cost of the blouse. Write an expression for the cost of the skirt. b + 15 Eric has rock and classical CDs in his car. The number of rock CDs is 3 more than the number of classical CDs. Let c represent the number of classical CDs. Write an expression for the number of rock CDs. 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. b − 4 Marcella has 6 fewer male cousins than female cousins. Let f represent the number of female cousins. Write an expression for the number of boy cousins. 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. 2n − 7 Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.

In the following exercises, use algebraic expressions to solve the problem. Car insurance Justin’s car insurance has a $750 deductible per incident. This means that he pays $750 and his insurance company will pay all costs beyond $750. If Justin files a claim for $2,100, how much will he pay, and how much will his insurance company pay? He will pay $750. His insurance company will pay $1350. Home insurance Pam and Armando’s home insurance has a $2,500 deductible per incident. This means that they pay $2,500 and their insurance company will pay all costs beyond $2,500. If Pam and Armando file a claim for $19,400, how much will they pay, and how much will their insurance company pay?

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. Explain the difference between “4 times the sum of x and y” and “the sum of 4 times x and y.”

ⓐ 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?

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.