2. Solving Equations Using Addition and Subtraction A.4f Apply these skills to solve practical problems. A.4b Justify steps used in solving equations. Use a graphing calculator to check your solutions. Objectives :

3. To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Ex: x = 5 is solved for x. y = 2x - 1 is solved for y.

4. Addition Property of Equality For any numbers a, b, and c, if a = b, then a + c = b + c. What it means: You can add any number to BOTH sides of an equation and the equation will still hold true.

5. An easy example: We all know that 7 = 7. Does 7 + 4 = 7? NO! But 7 + 4 = 7 + 4. The equation is still true if we add 4 to both sides. Would you ever leave the house with only one shoe on? Would you ever put blush on just one cheek? Would you ever shave just one side of your face?

6. Let’s try another example! x - 6 = 10 Add 6 to each side. x - 6 = 10 +6 +6 x = 16 Always check your solution!! The original problem is x - 6 = 10. Using the solution x=16, Does 16 - 6 = 10? YES! 10 = 10 and our solution is correct.

7. What if we see y + (-4) = 9? Recall that y + (-4) = 9 is the same as y - 4 = 9. Now we can use the addition property. y - 4 = 9 +4 +4 y = 13 Check your solution! Does 13 - 4 = 9? YES! 9=9 and our solution is correct.

8. How about -16 + z = 7? Remember to always use the sign in front of the number. Because 16 is negative, we need to add 16 to both sides. -16 + z = 7 +16 +16 z = 23 Check you solution! Does -16 + 23 = 7? YES! 7 = 7 and our solution is correct.

9. A trick question... -n - 10 = 5 +10 +10 -n = 15 Do we want -n? NO, we want positive n. If the opposite of n is positive 15, then n must be negative 15. Solution: n = -15 Check your solution! Does -(-15)-10=5? Remember, two negatives = a positive 15 - 10 = 5 so our solution is correct.

10. Subtraction Property of Equality For any numbers a, b, and c, if a = b, then a - c = b - c. What it means: You can subtract any number from BOTH sides of an equation and the equation will still hold true.

11. 3 Examples: 1) x + 3 = 17 -3 -3 x = 14 Does 14 + 3 = 17? 2) 13 + y = 20 -13 -13 y = 7 Does 13 + 7 = 20? 3) z - (-5) = -13 Change this equation. z + 5 = -13 -5 -5 z = -18 Does -18 -(-5) = -13? -18 + 5 = -13 -13 = -13 YES !

12. Try these on your own...

13. The answers...

14. Solving Equations Using Multiplication and Division A.4f Apply these skills to solve practical problems. A.4b Justify steps used in solving equations. Use a graphing calculator to check your solutions. Objectives :

15. Remember, To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Ex: x = 5 is solved for x. y = 2x - 1 is solved for y.

16. Multiplication Property of Equality For any numbers a, b, and c, if a = b, then ac = bc. What it means: You can multiply BOTH sides of an equation by any number and the equation will still hold true.

17. An easy example: We all know that 3 = 3. Does 3(4) = 3? NO! But 3(4) = 3(4). The equation is still true if we multiply both sides by 4. 4 Would you ever put deodorant under just one arm? 4 Would you ever put nail polish on just one hand? 4 Would you ever wear just one sock?

18. Let’s try another example! x = 4 2 Multiply each side by 2. (2)x = 4(2) 2 x = 8 Always check your solution!! The original problem is x = 4 2 Using the solution x = 8, Is x/2 = 4? YES! 4 = 4 and our solution is correct.

19. The two negatives will cancel each other out. The two fives will cancel each other out. (-5) x = -15 Does -(-15)/5 = 3? What do we do with negative fractions? Recall that Solve. Multiply both sides by -5.

20. Division Property of Equality 4 For any numbers a, b, and c (c ≠ 0), if a = b, then a/c = b/c What it means: 4 You can divide BOTH sides of an equation by any number - except zero- and the equation will still hold true. 4 Why did we add c ≠ 0?

21. 2 Examples: 1) 4x = 24 Divide both sides by 4. 4x = 24 4 4 x = 6 Does 4(6) = 24? YES! 2) -6x = 18 Divide both sides by -6. -6y = 18 -6 -6 y = -3 Does -6(-3) = 18? YES!

22. A fraction times a variable: The two step method: Ex: 2x = 4 3 1. Multiply by 3. (3)2x = 4(3) 3 2x = 12 2. Divide by 2. 2x = 12 2 2 x = 6 The one step method: Ex: 2x = 4 3 1. Multiply by the RECIPROCAL. (3)2x = 4(3) (2) 3 (2) x = 6

23. Try these on your own...

24. The answers...

25. Solving Multi-Step Equations Objectives: to solve equations involving more than one operation. to solve problems by working backward. A.4d Solving multistep linear equations. A.4f Solve real-world problems involving equations. A.4b Justify steps used in solving equations.

26. To Solve: Undo the operations by working backward. Ex: x + 9 = 6 5 Ask yourself: What is the first thing we are doing to x? The second thing? Recall the order of operations as you answer these questions. dividing by 5 adding 9 To undo these steps, do the opposite operations in opposite order.

27. The DO-UNDO chart Use a chart as a shortcut to answering the questions. DO UNDO ÷5 -9 +9 ·5 Follow the steps in the ‘undo’ column to isolate the variable. Ex: x + 9 = 6 5 First subtract 9. x + 9 - 9 = 6 - 9 5 x = -3 5 Then multiply by 5. (5) x = -3(5) 5 x = -15

28. Let’s try another! Complete the do-undo chart. DO UNDO -2 ·3 ÷ 3 +2 To solve for d: First multiply by 3. Then add 2. Ex: d - 2 = 7 3 (3) d - 2 = 7(3) 3 d - 2 = 21 +2 +2 d = 23

29. Here’s a tricky one! Remember to always use the sign in front of the number. DO UNDO ÷ -7 - 3 +3 · -7 To solve for a: First subtract 3. Then multiply by -7. Ex: 3 - a = -2 7 3 - a = -2 7 -3 -3 - a = -5 7 (-7)(- a) = (-5)(-7) 7 a = 35

30. Try a few on your own. 5z + 16 = 51 14n - 8 = 34 4b + 8 = 10 -2

31. The answers: DO UNDO · 5 - 16 +16 ÷ 5 z = 7 DO UNDO · 14 +8 -8 ÷ 14 n = 3 DO UNDO · 4 · -2 +8 - 8 ÷ -2 ÷ 4 b = -7

32. Solving Equations with the Variable on Both Sides Objectives: to solve equations with the variable on both sides. to solve equations containing grouping symbols.

33. To solve these equations, Use the addition or subtraction property to move all variables to one side of the equal sign.

34. Let’s see a few examples: 1) 6x - 3 = 2x + 13 Be sure to check your answer!

35. Let’s try another! 2) 3n + 1 = 7n - 5 Check:

36. Here’s a tricky one! 3) 5 + 2(y + 4) = 5(y - 3) + 10 Check: \

37. Let’s try one with fractions! 4) 3 - 2x = 4x - 6 3 = 6x - 6 9 = 6x so x = 3/2 Steps: Multiply each term by the least common denominator (8) to eliminate fractions. Solve for x. Add 2x. Add 6. Divide by 6.

38. Two special cases: 6(4 + y) - 3 = 4(y - 3) + 2y 24 + 6y - 3 = 4y - 12 + 2y 21 + 6y = 6y - 12 - 6y - 6y 21 = -12 Never true! 21 ≠ -12 NO SOLUTION! 3(a + 1) - 5 = 3a - 2 3a + 3 - 5 = 3a - 2 3a - 2 = 3a - 2 -3a -2 = -2 Always true! We write IDENTITY.

39. Try a few on your own: 9x + 7 = 3x - 5 8 - 2(y + 1) = -3y + 1 8 - 1 z = 1 z - 7 2 4

40. The answers: x = -2 y = -5 z = 20

41. A number is doubled and then increased by seven. The result is ninety-three. What is the original number?

42. Six less than five times a number is the same as seven times the number. What is the number?

43. Brad is a waiter, and he gets paid $5.75 per hour, and he can keep his tips. He knows his tips average $8.80 per table. If he worked an eight-hour shift and took home $169.20, how many tables did he serve?

44. In warmer climates, approximate temperature predictions can be made by counting the number of chirps a cricket makes during a minute. The temperature (in Fahrenheit) decreased by 40 is equivalent to one-forth of the number of cricket chirps in a minute. (a)Write an equation for this relationship. (b) Approximately how many chirps per minute should be recorded if the temperature is 90 F? (c) If a person recorded 48 cricket chirps in a minute, what would the temperature be?

45. Two trains leave a train station at the same time. One train travels east at 50 mph. The other train travels west at 55mph. In how many hours will the two trains be 315 miles apart?