5 When you are solving equations with integers, your goal is the same as with whole numbers: isolate the variable on one side of the equation.+–+–+–Recall that the sum of a number and its opposite is 0. When you add the opposite to get 0, you can isolate the variable.3 + (–3) = 0a + (–a) = 0
6 Additional Example 1A & 1B: Adding and Subtracting to Solve Equations Add 3 to both sides.x – = –3x = – 3B.–5 + r = 9–5 + r = 9– r = 9 + 5Add 5 to both sides.r = 14
7 The variable is already isolated. –6 + 8 = n 2 = n Add integers. Additional Example 1C & 1D: Adding and Subtracting to Solve Equations ContinuedSolve.C.–6 + 8 = nThe variable is already isolated.–6 + 8 = n2 = nAdd integers.D.z + 6 = –3z + 6 = –3–6Add –6 to each side.z = –9
8 Solve. A. B. p – 7 = – 9 p – 7 = – 9 p – 7 + 7 = – 9 + 7 Try This: Example 1A & 1BSolve.A.p – 7 = – 9p – 7 = – 9p – = – 9 + 7Add 7 to both sides.p – = –2p = –2B.–2 + g = 5–2 + g = 5– g = 5 + 2Add 2 to both sides.g = 7
9 The variable is already isolated. –1 + 7 = r Try This: Example 1C & 1DSolve.C.–1 + 7 = rThe variable is already isolated.–1 + 7 = r6 = rAdd integers.D.a + 9 = –9a + 9 = –9–9Subtract 9 from each side.a = –18
10 Solve. A. –5x = 35 –5x = 35 –5 Divide both sides by –5. x = –7 Additional Example 2A: Multiplying and Dividing to Solve Equations ContinuedSolve.A.–5x = 35–5x = 35–5Divide both sides by –5.x = –7
11 Multiply both sides by –4. Additional Example 2B: Multiplying and Dividing to Solve Equations ContinuedSolve.z –4B.= 5– = –4 5z –4Multiply both sides by –4.z = –20
12 Solve. A. –7x = 42 –7x = 42 Divide both sides by –7. –7 x = –6 Try This: Example 2ASolve.A.–7x = 42–7x = 42–7Divide both sides by –7.x = –6
13 Multiply both sides by –3. Try This: Example 2Solve.z –3B.= 9– = –3 9z –3Multiply both sides by –3.z = –27
14 Additional Example 3: Problem Solving Application Sarah heard on the morning news that the temperature had dropped 26 degrees since midnight. In the morning, the temperature was –8°F. What was the temperature at midnight?
15 Understand the Problem Additional Example 3 Continued1Understand the ProblemThe answer is the temperature at midnight.List the important information:The temperature dropped 26 degrees since midnight.In the morning it was –8°F.Show the relationship or the information:temperature at midnightmorning temperaturedegrees that temperature dropped–=
16 Additional Example 3 Continued 2Make a PlanWrite an equation and solve it. Let t represent the temperature at midnight and use the equation model.t – 26 = –8Solve3t – 26 = –8+ 26Add 26 to both sides.t = 18The temperature at midnight was 18°F.
17 Additional Example 3 Continued 4Look BackThe temperature at midnight was positive. Its value is less than the absolute value of the drop in temperature. This makes sense, since the morning temperature was negative.
18 Try This: Example 3Tim has two tug boats in a pulling contest. The boat on the left was pulling with a force of –23 newtons (N). If the net force is 5 N, what force is the boat on the right exerting on the rope?
19 Understand the Problem Try This: Example 3 Continued1Understand the ProblemThe answer is the force that the right tug boat is exerting.List the important information:The boat on the left pulls with a force of –23 N.The net force is 5 N.Show the relationship of the information:net forceleft boat’s force=right boat’s force+
20 Try This: Example 3 Continued 2Make a PlanWrite an equation and solve it. Let f represent the force in newtons.5 = –23 + fSolve35 = –23 + f+23+23Add 23 to both sides.28=fThe right boat is exerting a force of 28 newtons on the rope.–
21 Try This: Example 3 Continued Look Back4The left tug boat was pulling at a force of –23 N to the left. Since the total force on the tow rope was 5 N, the right tug boat was exerting a greater force in the positive direction. Thus the tug boat is moving to the right.
22 Lesson Quiz: Part 1Solve.1. t + 9 = –8t = –172. –15 = 3bb = –5x–4= –7x = 284. z – 16 = 30z = 46
23 Lesson Quiz: Part 25.A roller coaster descends down a hill at a rate of 80 feet per second. The bottom of the hill is 400 feet from the top. How long will it take the coaster riders to reach the bottom?5 seconds