1. Solve Equations with Rational Coefficients
Lesson 1
Solve Equations with Rational Coefficients

2. Multiplicative Inverse
Describe It
Define It
Two fractions that multiply to give 1.
The numerator and denominator of a fraction switches places.
List Some Examples
List Some Non-examples
1 2 𝑎𝑛𝑑 𝑎𝑛𝑑 6 5
𝑎𝑛𝑑 𝑎𝑛𝑑 − 1 5
Multiplicative Inverse

3. Inverse Property of Multiplication
Words: The product of a number and multiplicative inverse is 1.
Numbers: 𝑥 8 7 =1
Symbols: 𝑎 𝑏 ∙ 𝑏 𝑎 =1, 𝑤ℎ𝑒𝑟𝑒 𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑟𝑒 𝑛𝑜𝑡 0
What is the multiplicative inverse of ?
− 𝟐 𝟑
Inverse Property of Multiplication

4. coefficient
variable 5x
Coefficient

5. Example 1 Solve 3 4 c = 18. Check your solution. 3 4 c = 18
What’s the coefficient?
Multiply each side by the multiplicative inverse of
( 4 3 ) 3 4 c = 18( 4 3 )
1 c = 24
c = 24
3 4 (24) = 18? YES!
Example 1

6. Got it? Solve each equation. 1 5 x = 12 -24 = − 6 7 g
What’s the coefficient? 1/5
What’s the multiplicative inverse of 1/5?
Solution: x = 60
-24 = − 6 7 g
What’s the coefficient? -6/7
What’s the multiplicative inverse of -6/7? /6
Solution: g = 28
Got it?

7. Solve 𝑠= 𝑠= 𝑠= 33 2 ( 2 3 ) 3 2 𝑠= 33 2 ( 2 3 ) s = 11
What’s the coefficient?
3 2
What’s the multiplicative inverse?
2 3
Example 2

8. Got it? Solve the following equations. 4 1 6 𝑥=3 1 3 −1 7 8 𝑤=4 1 2
What’s the coefficient?
What’s the multiplicative inverse of ?
Solution: x = 1 1 4
−1 7 8 𝑤=4 1 2
What’s the coefficient?
What’s the multiplicative inverse of ? − 8 15
Solution: w =
Got it?

9. When an equation has a decimal, it is easier to divide then find the multiplicative inverse. Example: 10.8 = 0.9n What’s the coefficient? 0.9 Divide each side by = 0.9𝑛 = n
Dividing Decimals

10. Solve 3. 15 = 0. 45b 3. 15 = 0. 45b What’s the coefficient
Solve 3.15 = 0.45b 3.15 = 0.45b What’s the coefficient? 0.45 Divide each side by = 0.45𝑏 = b
Example 3

11. Latoya’s softball team won 75%, or 18 of its games. Define the variable.
Let g represent the number of games played.
Write an equation.
0.75g = 18
Solve the equation.
Divide each side by the coefficient.
g = 24
24 games were played.
Example 4

12. Solve Two-Step Equations
Lesson 2
Solve Two-Step Equations

13. Properties Addition Property of Equality Division Property of Equality
1 2 𝑥=10
2 ∙ 𝑥=10 ∙ 2
Division Property of Equality
3𝑥=9
3𝑥 3 = 9 3
Multiplication Property of Equality
x + 3 = 1
x + 3 – 3 = 1 - 3
Subtraction Property of Equality
x – 5 = 6
x – = 6 + 5
Properties

14. Solve 2x + 3 = 7. 2x + 3 = 7 Subtract 3 from each side
Solve 2x + 3 = 7. 2x + 3 = 7 Subtract 3 from each side. -3 = -3 2x = 4 What’s the coefficient? 2 Divide each side by the coefficient. x = 2
Example 1

15. Got it? Solve 5 + 2n = -1. 5 + 2n = -1 Subtract 5 from each side.
-5 = -5
2n = -6
What’s the coefficient? 2
Divide each side by the coefficient.
x = -3
Got it?

16. Solve 25 = 1 4 n – 3. 25 = 1 4 n – 3 Add 3 to each side
Solve 25 = 1 4 n – = 1 4 n – 3 Add 3 to each side. +3 = = 1 4 n What’s the coefficient? 1 4 Multiply each side by the multiplicative inverse. n = 112
Example 2

17. Solve -1 = 1 2 a + 9.
-1 = 1 2 a + 9
Subtract 9 from each side.
-1 = a + 9
- 9 = -9
-10 = 1 2 𝑎
What’s the coefficient?
Multiply each side by the multiplicative inverse.
a = -20
Got it?

18. Solve 6 – 3x = 21. 6 – 3x = 21 Subtract 6 from each side
Solve 6 – 3x = – 3x = 21 Subtract 6 from each side. - 6 = -6 -3x = 15 What’s the coefficient? -3 Divide each side by the coefficient. x = -5
Example 3

19. Solve -19 = -3x + 2. -19 = -3x + 2 Subtract 2 from each side
Solve -19 = -3x = -3x + 2 Subtract 2 from each side. - 2 = = -3x What’s the coefficient? -3 Divide each side by the coefficient. x = 7
Got it? 3

20. Chicago’s lowest recorded temperature in degrees Fahrenheit is -27
Chicago’s lowest recorded temperature in degrees Fahrenheit is -27. Solve the equation -27 = 1.8c + 32 to convert to degrees Celsius. -27 = 1.8c = = 1.8c Divide each side by = c Chicago’s lowest temperature was degrees Celsius.
Example 4

21. Write Two-Step Equations
Lesson 3
Write Two-Step Equations

22. Example 1 Translate each sentence into an equation.
Eight less than three times a number is -23.
Let “n” represent the number.
8 – 3n = -23
Thirteen is 7 more than one-fifth of a number.
Let “n” represent the number.
13 = n
Example 1

23. You buy 3 books that each cost the same amount and a magazine, all for $ You know that the magazine costs $1.99. How much does each book cost? Words : Three books and a magazine cost $ Variable: Let b represent the cost of 1 book. Equation: 3b = b = b = b = 18 Each book cost $18.
Example 2

24. A personal trainer buys a weight bench for $500 and w weights for $24
A personal trainer buys a weight bench for $500 and w weights for $24.99 each. The total cost of the purchase is $ How many weights were purchased? Words : Bench plus $24.99 per weight equals $ Variable: Let w represent the number of weights. Equation: w = $ w = $ w = w = 14 The personal trainer bought 14 weights.
Got it?

25. You and your friend’s lunch cost $19
You and your friend’s lunch cost $19. Your lunch cost $3 more than your friend’s. How much was your friend’s lunch? Words : Your friend’s lunch plus your lunch equals $19. Variable: Let f represent the cost of your friend’s lunch. Equation: f + (f + 3) = 19. f + (f + 3) = 19 2f + 3 = 19 2f = 16 f = 8 Your friend’s lunch cost $8.
Example 3

26. Write an equation for the following situation: An appliance repairman charges $35 for a house call and $30 per hour. The cost of the house call and the repair job came to $125.
Ticket Out The Door

27. Lesson 31
Collecting Like Terms

28. The coefficient of 3y is 3. The numerical coefficient of r is 1
The coefficient of 3y is 3. The numerical coefficient of r is 1. What is the numerical coefficient of –w ?
Adding Like Terms

30. 3a - a + 8b
2a + 8b
Adding Like Terms

31. a + 6m - 4b + b + 2a
3a – 3b + 6m
Adding Like Terms

32. m + 5m + 18c - 9c
9c + 6m
Adding Like Terms

33. 5s + 6t + t + 13s
18s + 7t
Adding Like Terms

35. Solve Equations with Variables on Both Sides
Lesson 4
Solve Equations with Variables on Both Sides

36. A wireless company offers two cell phone plans. Plan A charges $24
A wireless company offers two cell phone plans. Plan A charges $24.95 per month plus $0.10 per minute for calls. Plan B charges $19.95 per month plus $0.20 per minute. Use the questions to find the when the two plans cost the same.
Minutes (m)
Plan A m
Plan B m 30 40 50 60
For what value(s) do both Plans have the same cost?
50 minutes
27.95
25.95
28.95
27.95
29.95
29.95
Problem of the Day
30.95
31.95
Real World Example

37. Solve 8 + 4d = 5d d = 5d Subtract 4d from each side to combine like terms. -4d = -4d 8 = 1d d = 8
Example 1

38. Solve 6n – 1 = 4n – 5. 6n – 1 = 4n – 5 Subtract 4n from both sides
Solve 6n – 1 = 4n – 5. 6n – 1 = 4n – 5 Subtract 4n from both sides. -4n = -4n 2n – 1 = -5 Add 1 to each side. 2n = -4 n = -2
Example 2

39. Got it? Solve each equation. 8a = 5a + 21 3x – 7 = 8x + 23 a = 7

40. Green Gym chargers a one time fee of $50 plus $30 per session for a personal trainer. A new fitness center charges a yearly fee of $250 plus $10 for reach session with a trainer. For how many sessions is the cost of the two plans the same? $50 plus $30 a session equals $250 plus $10 per session s = s -10s = -10s s = s = 200 s = 10 The plans are the same when you attend 10 sessions.
Example 3

41. Solve 2 3 𝑥 – 1 = 𝑥. Make the fractions have the same denominator. 4 6 𝑥 – 1 = 𝑥 𝑥 = 𝑥 5 6 𝑥 – 1 = 9 +1 = 𝑥 = 10
( 6 5 ) 5 6 𝑥 =
x = 12
Example 4

42. Solve 1 2 p + 7 = 3 4 p + 9. p = -8
Got it?

43. Solve Multi-Step Equations
Lesson 5
Solve Multi-Step Equations

44. Distributive Property
This is when you distribute the outside number to all the numbers in the parentheses. Example: 8(5x + 4) 8(5x) + 8(4) 40x + 32
Distributive Property

45. Use the Distributive Property to solve
Use the Distributive Property to solve. 15(20 + d) = 420 Distribute the 15 to the 20 and d. 15(20) + 15(d) = d = = d = 120 Divide each side by 15. d = 8
Example 1

46. Got it? Use the Distributive Property to solve. -3(9 + x) = 33
5(a – 7) = 25
a = 12
Got it?

47. Number of Solutions Null Set One Solution Identity WORDS SYMBOLS
No solution
One solution
Infinitely many solutions
SYMBOLS
a = b
x = a
a = a
EXAMPLE
3x + 4 = 3x
4 = 0
Since 4  0, there is no solution.
2x = 20
x = 10
4x + 2 = 4x + 2
2 = 2
Since 2 = 2, the solution is all numbers.
Null Set: no solution and can be represented by { } or .
Identity: an equation that is true for any number.
Number of Solutions

48. Solve 6(x – 3) + 10 = 2(3x – 4). 6(x – 3) + 10 = 2(3x – 4) Distribute
Solve 6(x – 3) + 10 = 2(3x – 4). 6(x – 3) + 10 = 2(3x – 4) Distribute. 6x – = 6x – 8 Combine Like Terms 6x – 8 = 6x – 8 These equations are the same. Solution: Identity
Example 2

49. Solve. 8(4 – 2x) = 4(3 – 5x) + 4x. 8(4 – 2x) = 4(3 – 5x) + 4x Distribute. 32 – 16x = 12 – 20x + 4x Combine Like Terms 32 – 16x = 12 – 16x + 16x = +16x 32 = 12 This is false, so the solution is the null set or no solution.
Example 3

50. Got it? Solve each equation. 3(6 – 4x) = -2(6x – 9)
identify
2(3x + 5) = 5(2x – 4) – 4x
null set or no solution
Got it?

51. At a fair, Hunter bought 3 snacks and 10 ride tickets
At a fair, Hunter bought 3 snacks and 10 ride tickets. Each ride ticket costs $1.50 less than a snack. If he spent a total of $24.00, what was the cost of each snack? 3s + 10(s – 1.5) = 24 3s + 10s – 15 = 24 13s – 15 = = s = 39 s = 3 Each snack cost $3.
Example 4