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 an equation representing this scenario and solve it in multiple steps, including the combining of like terms, to determine how much each raffle ticket costs?

### Multi-step Equations with Like Terms

So far, you have learned how to solve one-step equations of the form \begin{align*}y=ax\end{align*} and two-step equations of the form \begin{align*}y = ax+b\end{align*}. This Concept will expand upon solving equations to include solving multi-step equations and equations involving the Distributive Property. The following are the general steps that you should take to solve an equation.

**Procedure to Solve Equations:**

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

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

**Step 3:** Isolate the \begin{align*}ax\end{align*} 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.

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

**Step 5:** Check your solution.

#### Let's solve the following problems:

- 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.

\begin{align*}3p + 4p + 37 = 79\end{align*}

This equation requires three steps to solve.

\begin{align*}3p + 4p + 37 = 79\end{align*}

Combine like terms: \begin{align*}7p+37=79.\end{align*}

Apply the Addition Property of Equality: \begin{align*}7p+37-37=79-37.\end{align*}

Simplify: \begin{align*}7p=42.\end{align*}

Apply the Multiplication Property of Equality: \begin{align*}7p \div 7=42 \div 7.\end{align*}

The solution is \begin{align*}p=6\end{align*}.

Checking the solution:

3 cans of soda per person for 6 people means that you should have 18 cans of soda. 4 slices of pizza per person for 6 people means that you should have 24 slices of pizza. Together, that is 42 sodas and slices. Adding the 37 party favors, we get 79 items. The answer is correct.

Six people are coming to the party.

- Kashmir needs to fence in his puppy. He will fence in three sides of a rectangle, 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?

Translate the sentence into an algebraic equation. Let \begin{align*}w\end{align*} represent the width of the enclosure.

\begin{align*}w + w + 12 = 40\end{align*}

Solve for \begin{align*}w\end{align*}.

\begin{align*}2w + 12 & = 40 \\ 2w + 12 - 12 & = 40-12 \\ 2w & = 28 \\ 2w \div 2 & = 28 \div 2 \\ w & = 14\end{align*}

Checking the solution:

For the 3 sided enclosure, there will be two sides that are the width. If the width is 14, then there will be 28 feet of fencing for those two sides. The length is 12 feet so adding that to the other two sides, we get 40 feet of fencing. The answer is correct.

The enclosure is 14 feet wide by 12 feet long.

- Solve for \begin{align*}v\end{align*} when \begin{align*}3v+5-7v+18=17.\end{align*}

\begin{align*}&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}\\ \end{align*}

Checking the solution:

\begin{align*}&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\end{align*}

### Examples

#### Example 1

Earlier, you were told that before today $96 worth of tickets had been sold and that today you sold 25 tickets and your classmate sold 35 tickets. $576 worth of tickets have been sold. How much does each raffle ticket cost?

To solve this problem, we need to write an equation to solve.

Let's represent the cost of the raffle ticket with the variable \begin{align*}c\end{align*}. The equation that represents this situation is: \begin{align*}96 + 25c + 35c = 576\end{align*}.

Solve for \begin{align*}c\end{align*}.

If you sold 25 tickets at $8 each, then you made $200.

If your friend sold 35 tickets at $8 each, then they made $280.

Together, you and your friend made $480. Adding that to the $96 worth of tickets that had been previously sold, $576 worth of tickets have been sold. The answer is correct.

Each ticket costs $8.

#### Example 2

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

\begin{align*} \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 \end{align*}

Checking the solution:

\begin{align*}5 (2w-\frac{3}{5})+10=w+16\\ 5 (2(1)-\frac{3}{5})+10=1+16\\ 5 (2-\frac{3}{5})+10=17\\ 10-3+10=17\\ 7+10=17\\ 17=17 \end{align*}

### Review

In 1 – 3, solve the equation.

- \begin{align*}f-1+2f+f-3=-4\end{align*}
- \begin{align*}2x+3+5x=18\end{align*}
- \begin{align*}5-7y-2y+10y=30\end{align*}

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

- Find four consecutive even integers whose sum is 244.
- Four more than two-thirds of a number is 22. What is the number?
- 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?
- Camden High has five times as many desktop computers as laptops. The school has 65 desktop computers. How many laptops does it have?
- 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?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.4.