8When 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
9Additional Example 1A & 1B: Adding and Subtracting to Solve Equations Add 3 to both sides.x = – 3x – = – 3Commutative PropertyB.–5 + r = 9–5 + r = 9– r = 9 + 5Add 5 to both sides.r = 14
10The 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
11Solve. 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 – = – 2Commutative PropertyB.–2 + g = 5–2 + g = 5– g = 5 + 2Add 2 to both sides.g = 7
12The 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 = –9–9Add –9 to each side.a = –18
13Solve. 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
14Multiply 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
15Solve. 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
16Multiply both sides by –3. Try This: Example 2Solve.z –3B.= 9– = –3 9z –3Multiply both sides by –3.z = –27
17Additional 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?
18Understand 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–=
19Additional 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.
20Additional 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.
21Try This: Example 3Tim has two tug boats in a pulling contest. The boat on the left was pulling with a force of 23 tons. If the net force is 50 tons, what force is the boat on the right exerting on the rope?
22Understand 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 tons.The net force is 50 tons.Show the relationship of the information:net forceleft boat’s force=right boat’s force+
23Try This: Example 3 Continued 2Make a PlanWrite an equation and solve it. Let f represent the force in tons.50 = 23 + fSolve350 = f–23–23Add –23 to both sides.27=fThe right boat is exerting a force of 27 tons.–
24Try This: Example 3 Continued Look Back4The left tug boat was pulling at a force of 23 tons to the left. The total force on the tow rope was 50 tons. Since 23 tons is less than half the force, the tug boat on the right was advancing forward in a positive direction while the left tug boat was moving backward in a negative direction.
25Lesson Quiz: Part 1Solve.1. t + 9 = –8t = –172. –15 = 3bb = –5x–4= –7x = 284. z – 16 = 30z = 46
26Lesson 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