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# Solving Linear Equations

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1. Solving Linear Equations

2. Bellwork Solve for x: • x – 5 = 57 • 3 – x = 10 • 4x + 28 = 68 • -3x – 7 = 14 • 4x – 9 + x = 16

3. Problem No. 1 1. Mica bought a CD for P25 and 8 blank videotapes. The total cost was P265. Find the cost of each blank videotape.

4. Equation: 25 + 8x = 265 Solution: 25 + 8x = 265 8x = 265 – 25 8x = 240 8 8 Answer: P30 each videotape

5. Problem No. 2 2. Juan bought 5 orchids in pots and P200 rose plant at a fund raiser. He spent a total of P750. Find the cost of each orchid.

6. Equation: 5x + 200 = 750 Solution: 5x + 200 = 750 5x = 750 – 200 5x = 550 5 5 x = 110 Answer: The price of each orchid is P110.

7. Error Analysis 10x + 4 = -2 10x = 2 10x = 2 10 10 x = 2 or 1 10 5 Correct Solution: 10x + 4 = -2 10x = -2 – 4 10x = -6 10x = -6 10 10 x = -3/5

8. Topic: Solving One Step Equations Addition Property of Equality – A property that states that we must add the same number on both sides of the equation to make it equal or balance. Example: x – 4 = 7 x – 4 + 4 = 7 + 4 x = 11

9. Subtraction Property of Equality – A property that states that we must subtract the same number on both sides of the equation to make it equal or balance. x + 2 = 8 x + 2 – 2 = 8 – 2 x = 6

10. Multiplication Property of Equality – A property that states that we must multiply the same number on both sides of the equation to make it equal or balance. ½ x = 8 2 (1/2 x) = 8 (2) x = 16

11. Division Property of Equality – A property that states that we must divide the same number on both sides of the equation to make it equal or balance. 6x = 36 6x = 36 6 6 x = 6

12. Topic: Solving Two-Step Equations In order to solve two-step linear equations you need to use properties of equality. Example No. 1 2x – 4 = 10 Given 2x = 10 + 4 Isolate term with x 2x = 14 Divide both sides by 2 2 2 x = 7

13. Example No. 2 3x + 8 = 14 Given 3x = 14 – 8 Isolate the term with x 3x = 6 Divide both sides by 3 3 3 x = 2 Checking: Substitute x = 2 in the equation 3x + 8 = 14 3(2) + 8 = 14 6 + 8 = 14 Therefore it is correct!

14. Topic: Solving Multi-Step Equations In order to solve multi-step equations you need to use your knowledge about distributive property and combining like terms. Example No. 1 2x + 3x – 5 = 35 Given 5x – 5 = 35 Add like terms 5x = 35 + 5 Add 5 on both sides 5x = 40 Divide both sides by 5 5 5 x = 8

15. Example No. 2 3 ( x – 4 ) = 24 Given 3x – 12 = 24 Use Distributive Property 3x = 24 + 12 Add 12 on both sides 3x = 36 Divide both sides by 3 3 3 Checking: Substitute x = 12 in the given eq. 3 ( x – 4 ) = 24 3 (12 – 4 ) = 24 3 (8) = 24 Therefore it is correct! x = 12

16. Topic: Solving Equations with Variables on Both Sides. To solve an equation with variables on both sides, you need to put all the variable terms on one side. Example No. 1 9x + 2 = 4x – 18 Given 9x – 4x = -18 – 2 Isolate variable terms 5x = -20 Divide both sides by 5 5 5 x = -4

17. Example No. 2 5x – 8 = - 2x + 6 Given 5x + 2x = 6 + 8 Isolate variable terms 7x = 14 Divide both sides by 7 7 7 Checking: Substitute x = 2 in the given eq. 5x – 8 = -2x + 6 5(2) – 8 = -2(2) + 6 10 – 8 = -4 + 6 2 = 2 Therefore it is correct! x = 2

18. Board Drill 4x – 10 + 6x = 100

19. Board Drill 7 ( x – 5 ) = 21

20. Board Drill 3x + x – 2 = 3

21. Board Drill -6x + 3x – 9 = 18

22. Board Drill 9x + 10 = 2x + 31

23. Board Drill -4x + 7 = 6x - 3

24. Board Drill 6x - 9 = x + 36

25. Board Drill 7x + 9 = 3x + 25

26. Board Drill 5x + 8 = 7x

27. Topic: Word Problems Involving Linear Equations A verbal problem or a mathematically-worded problem is a problem of mathematical nature stated in plain words, and which would involve mathematical calculation of some kind before it can be solved.

28. There are no set of rules or methods Which enable us to solve all kinds of Problems, because things must be Remembered in relation to different Types of problems. Note:

29. The following are the general strategies for problem solving. 1. Read the problem carefully. Be sure that you understand what the problem is all about. 2. Take note of what is asked in the problem.

30. 3. Represent the unknown by any variable and other unknowns in terms of the same variable according to the conditions of the problem. 4. Formulate the equation.

32. Examples Number Relation Problems The sum of two numbers is 36. One number is 3 less than twice the other number. What are the numbers

33. Representation: let x = the other number 2x – 3 = one number Equation: x + 2x – 3 = 36 Solution: 3x – 3 = 36 3x = 36 + 3 3x = 39 Answers: 13 and 23

34. Another Example Seven more than twice a number is four less than thrice the number. What is the number?

35. Example Consecutive Numbers Problems The sum of three consecutive numbers is 135. Find the numbers.

36. Representation: let x = 1st number x + 1 = 2nd number x + 2 = 3rd number Equation: x + x + 1 + x + 2 = 135 Solution: 3x = 135 – 3 3x = 132 Answers: 44, 45, and 46

37. Another Example Find three consecutive numbers whose sum is 60.

38. Example Age Problems Lherry is 3 times as old as Jane. In 4 years time, Lherry will be twice as old as Jane. How old is Jane?

39. Representation: Equation: 3x + 4 = 2 ( x + 4)

40. Solution: 3x + 4 = 2 (x + 4) 3x + 4 = 2x + 8 3x – 2x = 8 – 4 x = 4 Answer: Jane is 4 years old while Lherry is 12 years old.

41. Another Example Paulson is four times older than Maria. If the total of their ages is 60. How old is Maria?

42. Board Drill Thirty more than thrice a number is 45. What is the number?

43. Board Drill Ten more than twice a number is 100. What is the number?

44. Board Drill The sum of three consecutive numbers is 93. Find the numbers.

45. Board Drill Daisy, 38 years old, is 8 years more than three times as old as her son. How old is her son?

46. Board Drill Leah is three times older than Marvin. If the total of their ages is 68. How old is Marvin?

47. Board Drill A moving van rents for P1,250 a day plus P5 per kilometer. Mrs. Santos’ bill for a two day rental was P2000. How many kilometers did she drive?