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**2.3 Solving Multi-Step Equations:**

Term: A number, a variable or a product or quotient of numbers and variables that is added or subtracted in an algebraic expression. Combining like Terms: Simplify the terms with the same variable using arithmetic.

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GOAL:

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**Ex: What is the solution of 5 = 5m – 23 + 2m**

We solve multi-step equations by forming a series of simpler equations by first using combination of like terms: Ex: What is the solution of 5 = 5m – m 5 = 5m – m Same variable 5 = 5m + 2m – 23 5 = 7m – 23 Add like terms Inverse of subtraction 28 = 7m __ ___ Inverse of multiplication Don’t forget to check!! m = 4

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REAL-WORLD: A family buys airline tickets online. Each ticket costs $167. The family buys travel insurance with each ticket for $19 per ticket. The web site charges a fee of $16 for the entire purchase. The family is charged a total of $1132. How many tickets did the family buy?

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** 167x + 19x + 16 = 1132 SOLUTION: Using the given info we have:**

Each ticket (x) costs $ 167x Insurance for each ticket is $ 19x One time charge of $ +16 The family paid = $1132 x + 19x + 16 = 1132

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**x = 6 Tickets 167x + 19x + 16 = 1132 Like terms 186x + 16 = 1132**

Inverse of Add 186x = 1116 Like terms 186x /186 = 1116 /186 Inverse of mult. x = 6 Tickets

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**What is the solution of 11x – 8 – 6x = 22**

YOU TRY IT: What is the solution of 11x – 8 – 6x = 22

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**Solution: 11x – 8 – 6x = 22 Given 11x – 6x – 8 = 22 Same variable**

Add like terms + 8 = +8 Inverse of subtraction 5x = 30 Inverse of multiplication __ ___ x = 6 Don’t forget to check!!

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**Ex: What is the solution of – 8 (2x -1)= 36**

Solving equations with Distributive Property: Ex: What is the solution of – 8 (2x -1)= 36 – 8 (2x -1)= 36 – 8(2x) – 8(-1)= 36 Distributive – 8 – 16x + 8 = 36 Multiplication – 8 – 8 Inverse of addition – 16x = 28 _____ ____ – – 16 Inverse of multiplication x = - 28/16 - 7/4 Don’t forget to check!!

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YOU TRY IT: What is the solution of 18 = 3(2x – 6 )?

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**Solution: Given 18 = 3(2x – 6) 18 = 3(2x) + 3(-6) Distributive 3**

Multiplication Inverse of subtraction 36 = 6x __ ___ Inverse of multiplication x = 6 Don’t forget to check!!

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**Ex: What is the solution of 𝟑𝒙 𝟒 - 𝒙 𝟑 = 10?**

Solving equations with fractions: Ex: What is the solution of 𝟑𝒙 𝟒 - 𝒙 𝟑 = 10? 𝟑 𝟑 ∙ 𝟑𝒙 𝟒 - 𝒙 𝟑 ∙ 𝟒 𝟒 = 10 Common Denominator 𝟗𝒙 𝟏𝟐 𝟒𝒙 𝟏𝟐 = 10 Multiplication 𝟗𝒙−𝟒𝒙 𝟏𝟐 = 10 Same Denominator 𝟓𝒙 𝟏𝟐 = 10 Subtraction

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**Check: 𝟑(𝟐𝟒) 𝟒 - (𝟐𝟒) 𝟑 = 10 3(6) - 8 18 - 8 10 = 10**

12 ∙ 𝟓𝒙 𝟏𝟐 = 10 ∙ 12 Inverse of division 5𝒙 = 120 5𝒙 = 120 ___ ___ Inverse of Multiplication 𝒙 = 24 Don’t forget to check!! Check: 𝟑(𝟐𝟒) 𝟒 - (𝟐𝟒) 𝟑 = 3(6) 10 = 10

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**What is the solution of 𝟐𝒃 𝟓 + 𝟑𝒃 𝟒 = 3?**

YOU TRY IT: What is the solution of 𝟐𝒃 𝟓 + 𝟑𝒃 𝟒 = 3?

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**Solution: 𝟒 𝟒 ∙ 𝟐𝒃 𝟓 + 𝟑𝒃 𝟒 ∙ 𝟓 𝟓 = 3 Common Denominator**

𝟒 𝟒 ∙ 𝟐𝒃 𝟓 + 𝟑𝒃 𝟒 ∙ 𝟓 𝟓 = 3 Common Denominator 𝟖𝒃 𝟐𝟎 𝟏𝟓𝒃 𝟐𝟎 = 3 𝟖𝒃+𝟏𝟓𝒃 𝟐𝟎 = 3 Add the numerators 𝟐𝟑𝒃 𝟐𝟎 = 3 Opposite of dividing by 20

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**Solution: (20) 𝟐𝟑𝒃 𝟐𝟎 = 3(20) Multiply by 20 both sides 23b = 60**

(20) 𝟐𝟑𝒃 𝟐𝟎 = 3(20) Multiply by 20 both sides 23b = 60 Opposite of multiplying by 23 b = 𝟔𝟎 𝟐𝟑 Don’t forget to Check!

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Check: 𝟐𝒃 𝟓 + 𝟑𝒃 𝟒 = 3 𝟐𝒃 𝟓 + 𝟑𝒃 𝟒 = 3 b = 𝟔𝟎 𝟐𝟑 Plug in our answer 𝟐( 𝟔𝟎 𝟐𝟑 ) 𝟓 𝟑( 𝟔𝟎 𝟐𝟑 ) 𝟒 = 3 𝟏𝟐𝟎 𝟐𝟑 𝟓 + 𝟏𝟖𝟎 𝟐𝟑 𝟒 = 3 We never divide fractions, We keep, change and flip! 𝟏𝟐𝟎 𝟐𝟑 ( 𝟏 𝟓 ) + 𝟏𝟖𝟎 𝟐𝟑 ( 𝟏 𝟒 )= 3

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**Cross Cancel: Divide a numerator with a denominator (only on the same fraction)**

24 45 𝟏𝟐𝟎 𝟐𝟑 ( 𝟏 𝟓 ) + 𝟏𝟖𝟎 𝟐𝟑 ( 𝟏 𝟒 )= 3 𝟐𝟒 𝟐𝟑 + 𝟒𝟓 𝟐𝟑 = 3 Add the numerators: since our denominators are the same. 𝟐𝟒+𝟒𝟓 𝟐𝟑 = 3 Since 69/23 = 3 and 3 =3 we have gotten the right answer. 𝟔𝟗 𝟐𝟑 = 3

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**VIDEOS: Multi-Step Equations Multi-Step**

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Class Work: Pages: Problems: As many as you need to master the concepts.

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Step 1: Simplify Both Sides, if possible Distribute Combine like terms Step 2: Move the variable to one side Add or Subtract Like Term Step 3: Solve for.

Step 1: Simplify Both Sides, if possible Distribute Combine like terms Step 2: Move the variable to one side Add or Subtract Like Term Step 3: Solve for.

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