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

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 2 1 5 6 𝑎𝑛𝑑 6 5 2 1 3 𝑎𝑛𝑑 7 3 1 5 𝑎𝑛𝑑 − 1 5 Multiplicative Inverse

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

coefficient variable 5x Coefficient

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

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

Solve 1 1 2 𝑠=16 1 2 . 1 1 2 𝑠=16 1 2 3 2 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

Got it? Solve the following equations. 4 1 6 𝑥=3 1 3 −1 7 8 𝑤=4 1 2 What’s the coefficient? 4 1 6 What’s the multiplicative inverse of 4 1 6 ? 6 25 Solution: x = 1 1 4 −1 7 8 𝑤=4 1 2 What’s the coefficient? -1 7 8 What’s the multiplicative inverse of -1 7 8 ? − 8 15 Solution: w = - 2 2 5 Got it?

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 0.9. 10.8 0.9 = 0.9𝑛 0.9 12 = n Dividing Decimals

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 0.45. 3.15 0.45 = 0.45𝑏 0.45 7 = b Example 3

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

Solve Two-Step Equations Lesson 2 Solve Two-Step Equations

Properties Addition Property of Equality Division Property of Equality 1 2 𝑥=10 2 ∙ 1 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 – 5 + 5 = 6 + 5 Properties

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

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?

Solve 25 = 1 4 n – 3. 25 = 1 4 n – 3 Add 3 to each side Solve 25 = 1 4 n – 3. 25 = 1 4 n – 3 Add 3 to each side. +3 = +3 28 = 1 4 n What’s the coefficient? 1 4 Multiply each side by the multiplicative inverse. n = 112 Example 2

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

Solve 6 – 3x = 21. 6 – 3x = 21 Subtract 6 from each side Solve 6 – 3x = 21. 6 – 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

Solve -19 = -3x + 2. -19 = -3x + 2 Subtract 2 from each side Solve -19 = -3x + 2. -19 = -3x + 2 Subtract 2 from each side. - 2 = -2 -21 = -3x What’s the coefficient? -3 Divide each side by the coefficient. x = 7 Got it? 3

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 + 32 -32 = -32 -59 = 1.8c Divide each side by 1.8 -32.8 = c Chicago’s lowest temperature was -32.8 degrees Celsius. Example 4

Write Two-Step Equations Lesson 3 Write Two-Step Equations

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 = 7 + 1 5 n Example 1

You buy 3 books that each cost the same amount and a magazine, all for $55.99. You know that the magazine costs $1.99. How much does each book cost? Words : Three books and a magazine cost $55.99. Variable: Let b represent the cost of 1 book. Equation: 3b + 1.99 = 55.99 3b + 1.99 = 55.99 3b = 54.00 b = 18 Each book cost $18. Example 2

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 $849.56. How many weights were purchased? Words : Bench plus $24.99 per weight equals $849.56. Variable: Let w represent the number of weights. Equation: 500 + 24.99w = $849.56 500 + 24.99w = $849.56 24.99w = 349.56 w = 14 The personal trainer bought 14 weights. Got it?

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

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

Lesson 31 Collecting Like Terms

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

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

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

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

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

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

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 24.95 + 0.10m Plan B 19.95 + 0.20m 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

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

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

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

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 50 + 30s = 250 + 10s -10s = -10s 50 + 20s = 250 20s = 200 s = 10 The plans are the same when you attend 10 sessions. Example 3

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

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

Solve Multi-Step Equations Lesson 5 Solve Multi-Step Equations

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

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) = 420 300 + 15d = 420 -300 = -300 15d = 120 Divide each side by 15. d = 8 Example 1

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

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

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 – 18 + 10 = 6x – 8 Combine Like Terms 6x – 8 = 6x – 8 These equations are the same. Solution: Identity Example 2

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

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?

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 = 24 + 15 = +15 13s = 39 s = 3 Each snack cost $3. Example 4