m81281 Solve Equations Using Integers; The Division Property of Equality By the end of this section, you will be able to: Determine whether an integer is a solution of an equation Solve equations with integers using the Addition and Subtraction Properties of Equality Model the Division Property of Equality Solve equations using the Division Property of Equality Translate to an equation and solve 4208f187-08ad-402a-b95f-0a9a46582e52 Before you get started, take this readiness quiz. Evaluatex+4whenx=−4. If you missed this problem, review . 0 Solve:y−6=10. If you missed this problem, review . 16 Translate into an algebraic expression 5 less than x. If you missed this problem, review . x−5

In Solve Equations with the Subtraction and Addition Properties of Equality, we saw that a solution of an equation is a value of a variable that makes a true statement when substituted into that equation. In that section, we found solutions that were whole numbers. Now that we’ve worked with integers, we’ll find integer solutions to equations. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number or an integer.Substitute the number for the variable in the equation. Simplify the expressions on both sides of the equation. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution. Determine whether each of the following is a solution of 2x−5=−13: ⓐ x=4 ⓑ x=−4 ⓒ x=−9. ⓐ Substitute 4 for x in the equation to determine if it is true. Multiply. Subtract. Since x=4 does not result in a true equation, 4 is not a solution to the equation. ⓑ Substitute −4 for x in the equation to determine if it is true. Multiply. Subtract. Since x=−4 results in a true equation, −4 is a solution to the equation. ⓒ Substitute −9 for x in the equation to determine if it is true. Substitute −9 for x. Multiply. Subtract. Since x=−9 does not result in a true equation, −9 is not a solution to the equation. Determine whether each of the following is a solution of 2x−8=−14: ⓐ x=−11 ⓑ x=11 ⓒ x=−3 ⓐ no ⓑ no ⓒ yes Determine whether each of the following is a solution of 2y+3=−11: ⓐ y=4 ⓑ y=−4 ⓒ y=−7 ⓐ no ⓑ no ⓒ yes

In Solve Equations with the Subtraction and Addition Properties of Equality, we solved equations similar to the two shown here using the Subtraction and Addition Properties of Equality. Now we can use them again with integers. When you add or subtract the same quantity from both sides of an equation, you still have equality. Subtraction Property of Equality Addition Property of Equality For any numbersa,b,c,ifa=bthena−c=b−c. For any numbersa,b,c,ifa=bthena+c=b+c. Solve: y+9=5. Subtract 9 from each side to undo the addition. Simplify. Check the result by substituting −4 into the original equation. y+9=5 Substitute −4 for y −4+9=?5 5=5✓ Since y=−4 makes y+9=5 a true statement, we found the solution to this equation. Solve: y+11=7 −4 Solve: y+15=−4 −19 Solve: a−6=−8 Add 6 to each side to undo the subtraction. Simplify. Check the result by substituting −2 into the original equation: Substitute −2 for a The solution to a−6=−8 is −2. Since a=−2 makes a−6=−8 a true statement, we found the solution to this equation. Solve: a−2=−8 −6 Solve: n−4=−8 −4

All of the equations we have solved so far have been of the form x+a=b or x−a=b. We were able to isolate the variable by adding or subtracting the constant term. Now we’ll see how to solve equations that involve division. We will model an equation with envelopes and counters in .

Here, there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope? To determine the number, separate the counters on the right side into 2 groups of the same size. So 6 counters divided into 2 groups means there must be 3 counters in each group (since 6÷2=3). What equation models the situation shown in ? There are two envelopes, and each contains x counters. Together, the two envelopes must contain a total of 6 counters. So the equation that models the situation is 2x=6.

We can divide both sides of the equation by 2 as we did with the envelopes and counters. We found that each envelope contains 3 counters. Does this check? We know 2·3=6, so it works. Three counters in each of two envelopes does equal six. shows another example.

Now we have 3 identical envelopes and 12 counters. How many counters are in each envelope? We have to separate the 12 counters into 3 groups. Since 12÷3=4, there must be 4 counters in each envelope. See .

The equation that models the situation is 3x=12. We can divide both sides of the equation by 3. Does this check? It does because 3·4=12. Doing the Manipulative Mathematics activity “Division Property of Equality” will help you develop a better understanding of how to solve equations using the Division Property of Equality. Write an equation modeled by the envelopes and counters, and then solve it. There are 4 envelopes, or 4 unknown values, on the left that match the 8 counters on the right. Let’s call the unknown quantity in the envelopes x. Write the equation. Divide both sides by 4. Simplify. There are 2 counters in each envelope. Write the equation modeled by the envelopes and counters. Then solve it. 4x = 12; x = 3 Write the equation modeled by the envelopes and counters. Then solve it. 3x = 6; x = 2

The previous examples lead to the Division Property of Equality. When you divide both sides of an equation by any nonzero number, you still have equality. For any numbers a,b,c,and c≠0, If a=bthen ac=bc. Solve:7x=−49. To isolate x, we need to undo multiplication. Divide each side by 7. Simplify. Check the solution. 7x=−49 Substitute −7 for x. 7(−7)=?−49 −49=−49✓ Therefore, −7 is the solution to the equation. Solve: 8a=56 7 Solve: 11n=121 11 Solve: −3y=63. To isolate y, we need to undo the multiplication. Divide each side by −3. Simplify Check the solution. −3y=63 Substitute −21 for y. −3(−21)=?63 63=63✓ Since this is a true statement, y=−21 is the solution to the equation. Solve: −8p=96 −12 Solve: −12m=108 −9

In the past several examples, we were given an equation containing a variable. In the next few examples, we’ll have to first translate word sentences into equations with variables and then we will solve the equations. Translate and solve: five more than x is equal to −3. five more than x is equal to −3 Translate x+5=−3 Subtract 5 from both sides. x+5−5=−3−5 Simplify. x=−8 Check the answer by substituting it into the original equation. x+5=−3−8+5=?−3−3=−3✓ Translate and solve: Seven more than x is equal to −2. x + 7 = −2; x = −9 Translate and solve: Eleven more thanyis equal to 2. y + 11 = 2; y = −9 Translate and solve: the difference of n and 6 is −10. the difference of n and 6 is −10 Translate. n−6=−10 Add 6 to each side. n−6+6=−10+6 Simplify. n=−4 Check the answer by substituting it into the original equation. n−6=−10−4−6=?−10−10=−10✓ Translate and solve: The difference of p and 2 is −4. p − 2 = −4; p = −2 Translate and solve: The difference of q and 7 is −3. q − 7 = −3; q = 4 Translate and solve: the number 108 is the product of −9 and y. the number of 108 is the product of −9 and y Translate. 108=−9y Divide by −9. 108−9=−9y−9 Simplify. −12=y Check the answer by substituting it into the original equation. 108=−9y108=?−9(−12)108=108✓ Translate and solve: The number 132 is the product of −12 and y. 132 = −12y; y = −11 Translate and solve: The number 117 is the product of −13 and z. 117 = −13z; z = −9 One-Step Equations With Adding Or Subtracting One-Step Equations With Multiplying Or Dividing

How to determine whether a number is a solution to an equation. Step 1. Substitute the number for the variable in the equation. Step 2. Simplify the expressions on both sides of the equation. Step 3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution. Properties of Equalities Subtraction Property of Equality Addition Property of Equality For any numbersa,b,c,ifa=bthena−c=b−c. For any numbersa,b,c,ifa=bthena+c=b+c. Division Property of Equality For any numbers a,b,c, and c≠0 If a=b, then ac=bc.

Determine Whether a Number is a Solution of an Equation In the following exercises, determine whether each number is a solution of the given equation. 4x−2=6 ⓐ x=−2 ⓑ x=−1 ⓒ x=2 ⓐ no ⓑ no ⓒ yes 4y−10=−14 ⓐ y=−6 ⓑ y=−1 ⓒ y=1 9a+27=−63 ⓐ a=6 ⓑ a=−6 ⓒ a=−10 ⓐ noⓑ no ⓒ yes 7c+42=−56 ⓐ c=2 ⓑ c=−2 ⓒ c=−14 Solve Equations Using the Addition and Subtraction Properties of Equality In the following exercises, solve for the unknown. n+12=5 n = −7 m+16=2 p+9=−8 p = −17 q+5=−6 u−3=−7 u = −4 v−7=−8 h−10=−4 h = 6 k−9=−5 x+(−2)=−18 x = −16 y+(−3)=−10 r−(−5)=−9 r = −14 s−(−2)=−11 Model the Division Property of Equality In the following exercises, write the equation modeled by the envelopes and counters and then solve it. 3x = 6; x = 2 2x = 8; x = 4 Solve Equations Using the Division Property of Equality In the following exercises, solve each equation using the division property of equality and check the solution. 5x=45 x = 9 4p=64 −7c=56 c = −8 −9x=54 −14p=−42 p = 3 −8m=−40 −120=10q q = −12 −75=15y 24x=480 x = 20 18n=540 −3z=0 z = 0 4u=0 Translate to an Equation and Solve In the following exercises, translate and solve. Four more than n is equal to 1. n + 4 = 1; n = −3 Nine more than m is equal to 5. The sum of eight and p is −3. 8 + p = −3; p = −11 The sum of two and q is −7. The difference of a and three is −14. a − 3 = −14; a = −11 The difference of b and 5 is −2. The number −42 is the product of −7 and x. −42 = −7x; x = 6 The number −54 is the product of −9 and y. The product of -15 and f is 75. −15f = 75; f = −5 The product of −18 and g is 36. −6 plus c is equal to 4. −6 + c = 4; c = 10 −2 plus d is equal to 1. Nine less than m is −4. m − 9 = −4; m = 5 Thirteen less than n is −10. Mixed Practice In the following exercises, solve. ⓐ x+2=10 ⓑ 2x=10 ⓐ x = 8ⓑ x = 5 ⓐ y+6=12 ⓑ 6y=12 ⓐ −3p=27 ⓑ p−3=27 ⓐ p = −9ⓑ p = 30 ⓐ −2q=34 ⓑ q−2=34 a−4=16 a = 20 b−1=11 −8m=−56 m = 7 −6n=−48 −39=u+13 u = −52 −100=v+25 11r=−99 r = −9 15s=−300 100=20d d = 5 250=25n −49=x−7 x = −42 64=y−4

Cookie packaging A package of 51 cookies has 3 equal rows of cookies. Find the number of cookies in each row, c, by solving the equation 3c=51. 17 cookies Kindergarten class Connie’s kindergarten class has 24 children. She wants them to get into 4 equal groups. Find the number of children in each group, g, by solving the equation 4g=24.

Is modeling the Division Property of Equality with envelopes and counters helpful to understanding how to solve the equation 3x=15? Explain why or why not. Sample answer: It is helpful because it shows how the counters can be divided among the envelopes. Suppose you are using envelopes and counters to model solving the equations x+4=12 and 4x=12. Explain how you would solve each equation. Frida started to solve the equation −3x=36 by adding 3 to both sides. Explain why Frida’s method will not solve the equation. Sample answer: The operation used in the equation is multiplication. The inverse of multiplication is division, not addition. Raoul started to solve the equation 4y=40 by subtracting 4 from both sides. Explain why Raoul’s method will not solve the equation.

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

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

Locate Positive and Negative Numbers on the Number Line In the following exercises, locate and label the integer on the number line. 5 −5 −3 3 −8 −7 Order Positive and Negative Numbers In the following exercises, order each of the following pairs of numbers, using < or >. 4__8 < −6__3 −5__−10 > −9__−4 2__−7 > −3__1 Find Opposites In the following exercises, find the opposite of each number. 6 −6 −2 −4 4 3 In the following exercises, simplify. ⓐ −(8) ⓑ −(−8) ⓐ −8 ⓑ 8 ⓐ −(9) ⓑ −(−9) In the following exercises, evaluate. −x,when ⓐ x=32 ⓑ x=−32 ⓐ −32 ⓑ 32 −n,when ⓐ n=20 ⓐ n=−20 Simplify Absolute Values In the following exercises, simplify. |−21| 21 |−42| |36| 36 −|15| |0| 0 −|−75| In the following exercises, evaluate. |x|whenx=−14 14 −|r|whenr=27 −|−y|wheny=33 −33 |−n|whenn=−4 In the following exercises, fill in <,>,or= for each of the following pairs of numbers. −|−4|__4 < −2__|−2| −|−6|__−6 = −|−9|__|−9| In the following exercises, simplify. −(−55)and−|−55| 55; −55 −(−48)and−|−48| |12−5| 7 |9+7| 6|−9| 54 |14−8|−|−2| |9−3|−|5−12| −1 5+4|15−3| Translate Phrases to Expressions with Integers In the following exercises, translate each of the following phrases into expressions with positive or negative numbers. the opposite of 16 −16 the opposite of −8 negative 3 −3 19 minus negative 12 a temperature of 10 below zero −10° an elevation of 85 feet below sea level

Model Addition of Integers In the following exercises, model the following to find the sum. 3+7 10 −2+6 5+(−4) 1 −3+(−6) Simplify Expressions with Integers In the following exercises, simplify each expression. 14+82 96 −33+(−67) −75+25 −50 54+(−28) 11+(−15)+3 −1 −19+(−42)+12 −3+6(−1+5) 21 10+4(−3+7) Evaluate Variable Expressions with Integers In the following exercises, evaluate each expression. n+4when ⓐ n=−1 ⓑ n=−20 ⓐ 3 ⓑ −16 x+(−9)when ⓐ x=3 ⓑ x=−3 (x+y)3whenx=−4,y=1 −27 (u+v)2whenu=−4,v=11 Translate Word Phrases to Algebraic Expressions In the following exercises, translate each phrase into an algebraic expression and then simplify. the sum of −8 and 2 −8 + 2 = −6 4 more than −12 10 more than the sum of −5 and −6 10 + [−5 + (−6)] = −1 the sum of3and−5,increased by 18 Add Integers in Applications In the following exercises, solve. Temperature On Monday, the high temperature in Denver was −4 degrees. Tuesday’s high temperature was 20 degrees more. What was the high temperature on Tuesday? 16 degrees Credit Frida owed $75 on her credit card. Then she charged $21 more. What was her new balance?

Model Subtraction of Integers In the following exercises, model the following. 6−1 5 −4−(−3) 2−(−5) 7 −1−4 Simplify Expressions with Integers In the following exercises, simplify each expression. 24−16 8 19−(−9) −31−7 −38 −40−(−11) −52−(−17)−23 −58 25−(−3−9) (1−7)−(3−8) −1 32−72 Evaluate Variable Expressions with Integers In the following exercises, evaluate each expression. x−7when ⓐ x=5 ⓑ x=−4 ⓐ −2 ⓑ −11 10−ywhen ⓐ y=15 ⓑ y=−16 2n2−n+5whenn=−4 41 −15−3u2whenu=−5 Translate Phrases to Algebraic Expressions In the following exercises, translate each phrase into an algebraic expression and then simplify. the difference of −12and5 −12 − 5 = −17 subtract 23 from −50 Subtract Integers in Applications In the following exercises, solve the given applications. Temperature One morning the temperature in Bangor, Maine was 18 degrees. By afternoon, it had dropped 20 degrees. What was the afternoon temperature? −2 degrees Temperature On January 4, the high temperature in Laredo, Texas was 78 degrees, and the high in Houlton, Maine was −28degrees. What was the difference in temperature of Laredo and Houlton?

Multiply Integers In the following exercises, multiply. −9⋅4 −36 5(−7) (−11)(−11) 121 −1⋅6 Divide Integers In the following exercises, divide. 56÷(−8) −7 −120÷(−6) −96÷12 −8 96÷(−16) 45÷(−1) −45 −162÷(−1) Simplify Expressions with Integers In the following exercises, simplify each expression. 5(−9)−3(−12) −9 (−2)5 −34 −81 (−3)(4)(−5)(−6) 42−4(6−9) 54 (8−15)(9−3) −2(−18)÷9 4 45÷(−3)−12 Evaluate Variable Expressions with Integers In the following exercises, evaluate each expression. 7x−3whenx=−9 −66 16−2nwhenn=−8 5a+8bwhena=−2,b=−6 −58 x2+5x+4whenx=−3 Translate Word Phrases to Algebraic Expressions In the following exercises, translate to an algebraic expression and simplify if possible. the product of −12 and 6 −12(6) = −72 the quotient of 3 and the sum of −7 and s

Determine Whether a Number is a Solution of an Equation In the following exercises, determine whether each number is a solution of the given equation. 5x−10=−35 ⓐ x=−9 ⓑ x=−5 ⓒ x=5 ⓐ no ⓑ yes ⓒ no 8u+24=−32 ⓐ u=−7 ⓑ u=−1 ⓒ u=7 Using the Addition and Subtraction Properties of Equality In the following exercises, solve. a+14=2 −12 b−9=−15 c+(−10)=−17 −7 d−(−6)=−26 Model the Division Property of Equality In the following exercises, write the equation modeled by the envelopes and counters. Then solve it. 3x = 9; x = 3 Solve Equations Using the Division Property of Equality In the following exercises, solve each equation using the division property of equality and check the solution. 8p=72 9 −12q=48 −16r=−64 4 −5s=−100 Translate to an Equation and Solve. In the following exercises, translate and solve. The product of −6 andyis−42 −6y = −42; y = 7 The difference ofzand −13 is −18. Four more than m is −48. m + 4 = −48; m = −52 The product of −21 andnis 63.

Describe how you have used two topics from this chapter in your life outside of your math class during the past month. Answers will vary.

Locate and label 0,2,−4, and −1 on a number line. In the following exercises, compare the numbers, using <or>or=. ⓐ −6__3 ⓑ −1__−4 ⓐ < ⓑ > ⓐ −5__|−5| ⓑ −|−2|__−2 In the following exercises, find the opposite of each number. ⓐ −7 ⓑ 8 ⓐ 7ⓑ −8 In the following exercises, simplify. −(−22) |4−9| 5 −8+6 −15+(−12) −27 −7−(−3) 10−(5−6) 11 −3⋅8 −6(−9) 54 70÷(−7) (−2)3 −8 −42 16−3(5−7) 22 |21−6|−|−8| In the following exercises, evaluate. 35−awhena=−4 39 (−2r)2whenr=3 3m−2nwhenm=6,n=−8 34 −|−y|wheny=17 In the following exercises, translate each phrase into an algebraic expression and then simplify, if possible. the difference of −7 and −4 −7 − (−4) = −3 the quotient of 25 and the sum of m and n. In the following exercises, solve. Early one morning, the temperature in Syracuse was −8°F. By noon, it had risen 12°. What was the temperature at noon? 4°F Collette owed $128 on her credit card. Then she charged $65. What was her new balance? In the following exercises, solve. n+6=5 n = −1 p−11=−4 −9r=−54 r = 6 In the following exercises, translate and solve. The product of 15 andxis 75. Eight less thanyis −32. y − 8 = −32; y = −24