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Multi-Step Equations with Like Terms

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Suppose you and your classmate are selling raffle tickets. Before today, $96 worth of tickets had been sold, and today, you sold 25 tickets and your classmate sold 35 tickets. Currently, $576 worth of tickets have been sold. Can you write and equation representing this scenario and solve it in multiple steps, including the combining of like terms, to determine how much each raffle ticket costs? In this Concept, you'll learn how to solve these types of problems.

Guidance

So far, you have learned how to solve one-step equations of the form y=ax and two-step equations of the form y = ax+b . This Concept will expand upon solving equations to include solving multi-step equations and equations involving the Distributive Property.

Solving Multi-Step Equations by Combining Like Terms

In the last Concept, you learned the definition of like terms and how to combine such terms. We will use the following situation to further demonstrate solving equations involving like terms.

You are hosting a Halloween party. You will need to provide 3 cans of soda per person, 4 slices of pizza per person, and 37 party favors. You have a total of 79 items. How many people are coming to your party?

This situation has several pieces of information: soda cans, slices of pizza, and party favors. Translate this into an algebraic equation.

3p + 4p + 37 = 79

This equation requires three steps to solve. In general, to solve any equation you should follow this procedure.

Procedure to Solve Equations:

1. Remove any parentheses by using the Distributive Property or the Multiplication Property of Equality.

2. Simplify each side of the equation by combining like terms.

3. Isolate the ax term. Use the Addition Property of Equality to get the variable on one side of the equal sign and the numerical values on the other.

4. Isolate the variable. Use the Multiplication Property of Equality to get the variable alone on one side of the equation.

5. Check your solution.

Example A

Determine the number of party-goers in the opening example.

Solution: 3p + 4p + 37 = 79

Combine like terms: 7p+37=79.

Apply the Addition Property of Equality: 7p+37-37=79-37.

Simplify: 7p=42.

Apply the Multiplication Property of Equality: 7p \div 7=42 \div 7.

The solution is p=6 .

There are six people coming to the party.

Example B

Kashmir needs to fence in his puppy. He will fence in three sides, connecting it to his back porch. He wants the length to be 12 feet, and he has 40 feet of fencing. How wide can Kashmir make his puppy enclosure?

Solution: Translate the sentence into an algebraic equation. Let w represent the width of the enclosure.

w + w + 12 = 40

Solve for w .

2w + 12 & = 40 \\2w + 12 - 12 & = 40-12 \\2w & = 28 \\2w \div 2 & = 28 \div 2 \\w & = 14

The dimensions of the enclosure are 14 feet wide by 12 feet long.

Example C

Solve for v when 3v+5-7v+18=17.

Solution:

&3v+5-7v+18=17\\&3v-7v+18+5=17\\&-4v+23=17\\&-4v+23-23=17-23\\&-4v=-6\\&-\frac{1}{4}\cdot -4v=-\frac{1}{4}\cdot-6\\& v=\frac{6}{4}\\& v=\frac{3}{2}\\

Checking the solution:

&3\cdot \frac{3}{2}+5-7\cdot \frac{3}{2}+18=17\\&\frac{9}{2}+5-\frac{21}{2}+18=17\\&-\frac{12}{2}+23=17\\&-6+23=17\\&17=17

Video Review

Guided Practice

Solve for w when 5\left(2w-\frac{3}{5}\right)+10=w+16 .

Solution:

 \text{Start by distributing the 5.} && 5\left(2w-\frac{3}{5}\right)+10&=w+16\\&&  \Rightarrow 10w-3+10&=w+16\\\text{Combine like terms on the left side.} && \Rightarrow 10w+7&=w+16 \\\text{Subtract 7 and }w \text{ from each side.} && \Rightarrow 10w+7-7-w&=w+16-7-w\\&& \Rightarrow 9w&=9\\\text{Isolate }w \text{ by dividing each side by 9.} && \Rightarrow \frac{9w}{9}&=\frac{9}{9}\\&& \Rightarrow w&=1

Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Multi-Step Equations (15:01)

In 1 – 3, solve the equation.

  1. f-1+2f+f-3=-4
  2. 2x+3+5x=18
  3. 5-7y-2y+10y=30

In 4-8, write an equation and then solve for the variable.

  1. Find four consecutive even integers whose sum is 244.
  2. Four more than two-thirds of a number is 22. What is the number?
  3. The total cost of lunch is $3.50, consisting of a juice, a sandwich, and a pear. The juice cost 1.5 times as much as the pear. The sandwich costs $1.40 more than the pear. What is the price of the pear?
  4. Camden High has five times as many desktop computers as laptops. The school has 65 desktop computers. How many laptops does it have?
  5. A realtor receives a commission of $7.00 for every $100 of a home’s selling price. How much was the selling price of a home if the realtor earned $5,389.12 in commission?

Mixed Review

  1. Simplify 1 \frac{6}{7} \times \frac{2}{3} .
  2. Define evaluate .
  3. Simplify \sqrt{75} .
  4. Solve for m: \frac{1}{9} m=12 .
  5. Evaluate: ((-5) - (-7) - (-3)) \times (-10) .
  6. Subtract: 0.125- \frac{1}{5} .

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