2. SOLVING ONE-VARIABLE EQUATIONS •. Goal: Find the one value
SOLVING ONE-VARIABLE EQUATIONS • Goal: Find the one value of the variable that makes the sentence true.

3. •. We can solve equations by. doing the OPPOSITE of what
• We can solve equations by doing the OPPOSITE of what has been done to the variable in the problem. • If a problem says +, you subtract. • If a problem has multiplication, you divide.

4. By doing the opposite, we keep the sides of the equation balanced.

5. 5x – 13 = 52

6. 5x – 13 = x = 65

7. 5x – 13 = x = x = 13

8. 12x = 2127

9. 12x = x = 27.75

10. 963 – 25x = 704

11. 963 – 25x = x = 10.36

12. What about this? 𝑦 5 −13=2

13. What about this? 𝑦 5 −13=2 Fractions mean division, so to cancel, we’ll add 13 and then multiply by  n = 75

14. Things that can complicate solving equations …

15. Parentheses •. Use distributive property. first. Like terms •
Parentheses • Use distributive property first. Like terms • Combine them first.

16. 4(3x – 7) = 48

17. 4(3x – 7) = x – 28 = 48

18. 4(3x – 7) = x – 28 = 48 12x = x = 6.333…

19. -7(2x – 11) = 98

20. -7(2x – 11) = 98 -14x + 77 = 98 -14x = 21 x = -3/2 or -1.5

21. 4p + 3 – 2p p + 2 = 17

22. 4p + 3 – 2p p + 2 = p + 12 = p = p = 5/7

23. 5(3x + 5) – 3(2x – 1) = 145

24. 5(3x + 5) – 3(2x – 1) = x + 25 – 6x + 3 = 145

25. 5(3x + 5) – 3(2x – 1) = x + 25 – 6x + 3 = x + 28 = 145

26. 5(3x + 5) – 3(2x – 1) = 145 15x + 25 – 6x + 3 = 145 9x + 28 = 145 9x = 117 x = 13

27. The goal is always to simplify
The goal is always to simplify. Make the problem look like the easy ones we know how to solve.

28. Variable on Both Sides • Find the smaller number of the variable, and subtract that on both sides. • Solve the remaining problem.

29. 5x – 15 = 2x + 72

30. 5x – 15 = 2x x x 3x – 15 =

31. 7x – 15 = 2x x x 3x – 15 = x = x =

32. 5x + 13 = 7x + 40

33. 5x + 13 = 7x x x = 2x + 40

34. 5x + 13 = 7x x x = 2x x = -27/2 or -13.5

35. 3(2x + 7) = 3x x + 9

36. 3(2x + 7) = 3x x x + 21 = 4x + 13

37. 3(2x + 7) = 3x + 4 + x + 9 6x + 21 = 4x + 13 2x + 21 = 13 2x = -8 x = -4

38. Special equations 2(3x – 7) = 6x + 11 10x – 15 = 5(2x – 3)

39. 2(3x – 7) = 6x x – 14 = 6x ?????

40. 10x – 15 = 5(2x – 3) 10x – 15 = 10x – ?????

41. When variables cancel out… •. If you have the exact
When variables cancel out… • If you have the exact same thing on both sides (like 8 = 8), the answer is ALL REAL NUMBERS or INFINITELY MANY SOLUTIONS. 10x – 15 = 5(2x – 3) 10x – 15 = 10x – = -15

42. An equation with infinitely many solutions can also be called an IDENTITY.

43. •. If there is something. different on the 2 sides
• If there is something different on the 2 sides (like 5 = 7), there is NO SOLUTION. 2(3x – 7) = 6x x – 14 = 6x = 11

44. •. If there is something. different on the 2 sides
• If there is something different on the 2 sides (like 5 = 7), there is NO SOLUTION. 2(3x – 7) = 6x x – 14 = 6x = 11