Suppose you are saving up to buy a leather jacket that costs $250. You currently have $90 put away, and you're saving at a rate of $40 per month. Can you write an equation to represent this situation and then solve it in two steps to find the number of months needed to reach your goal?

### Two Step Equations and Properties of Equality

Suppose Shaun weighs 146 pounds and wants to lose enough weight to wrestle in the 130-pound class. His nutritionist designed a diet for Shaun so he will lose about 2 pounds per week. How many weeks will it take Shaun to weigh enough to wrestle in his class?

This is an example that can be solved by working backward. In fact, you may have already found the answer by using this method. The solution is 8 weeks.

By translating this situation into an algebraic sentence, we can begin the process of solving equations. Recall that to solve an equation means to “undo” all the operations of the sentence, leaving a value for the variable.

Translate Shaun’s situation into an equation:

\begin{align*}-2w + 146 = 130\end{align*}

This sentence has two operations: addition and multiplication. To find the value of the variable, we must use both properties of equality: the Addition Property of Equality and the Multiplication Property of Equality.

To solve an equation of the form \begin{align*}ax + b =\end{align*} some number:

**Step 1:** Use the Addition Property of Equality to get the variable term \begin{align*}ax\end{align*} alone on one side of the equation:

\begin{align*}ax = some\ number\end{align*}

**Step 2:** Use the Multiplication Property of Equality to get the variable \begin{align*}x\end{align*} alone on one side of the equation:

\begin{align*}x = some\ number\end{align*}

#### Let's solve Shaun's problem:

Solve Shaun’s problem.

\begin{align*}-2 w + 146 = 130 \end{align*}

Apply the Addition Property of Equality: \begin{align*}-2w + 146 - 146 = 130-146.\end{align*}

Simplify: \begin{align*}-2w = -16.\end{align*}

Apply the Multiplication Property of Equality: \begin{align*}-2w \div -2 = -16 \div -2.\end{align*}

The solution is \begin{align*}w = 8\end{align*}.

It will take 8 weeks for Shaun to weigh 130 pounds.

#### Solving Equations by Combining Like Terms

Michigan has a 6% sales tax. Suppose you made a purchase and paid $95.12, including tax. How much was the purchase before tax?

Begin by determining the noun that is unknown and choose a letter as its representation.

The purchase price is unknown so this is our variable. Call it \begin{align*}p\end{align*}. Now translate the sentence into an algebraic equation.

\begin{align*}price + (0.06) price & = total\ amount \\ p + 0.06p & = 95.12\end{align*}

To solve this equation, you must know how to **combine like terms.**

**Like terms** are expressions that have **identical** variable parts.

According to this definition, you can only combine like terms if they are identical. Combining like terms only applies to addition and subtraction. This is not a true statement when referring to multiplication and division.

The numerical part of an algebraic term is called the **coefficient**. To combine like terms, you add (or subtract) the coefficients of the identical variable parts.

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

- Identify the like terms, and then combine.

\begin{align*}10b + 7bc + 4c + (-8b)\end{align*}

Like terms have identical variable parts. The only terms having identical variable parts are \begin{align*}10b\end{align*} and \begin{align*}-8b\end{align*}. To combine these like terms, add them together.

\begin{align*}10b + 7bc + 4c + -8b = 2b + 7bc + 4c\end{align*}

- \begin{align*}p + 0.06p = 95.12\end{align*}

Combine the like terms: \begin{align*}p + 0.06p = 1.06p\end{align*}, since \begin{align*}p = 1p.\end{align*}

Simplify: \begin{align*}1.06p = 95.12.\end{align*}

Apply the Multiplication Property of Equality: \begin{align*}1.06p \div 1.06 = 95.12 \div 1.06.\end{align*}

Simplify: \begin{align*}p = 89.74.\end{align*}

This is the solution to the question about Michigan's sales tax from above. Therefore the price before tax was $89.74.

### Examples

#### Example 1

Earlier, you were told that you are saving to buy a jacket that costs $250 and currently have $90 put away. If you are saving at a rate of $40 per month, how many months will it take to reach your goal?

Let \begin{align*}m\end{align*} represent the unknown number of months until you reach your goal. Your goal is $250 so that will the be number that the equation is set equal to and since you are starting with $90, that will be the constant number on the other side. If you are saving at a rate of $40 per month, \begin{align*}40m\end{align*} represents the amount that your savings is increasing each month.

The equation that represents this situation is: \begin{align*}250=90+40m\end{align*}

Apply the Addition Property of Equality and simplify:

Apply the Multiplication Property of Equality and simplify:

\begin{align*}160=40m\\ 160\div40=40m\div40\\ 4=m\end{align*}

It will take you 4 months to save up for the jacket.

#### Example 2

An emergency plumber charges $65 as a call-out fee plus an additional $75 per hour. He arrives at a house at 9:30 and works to repair a water tank. If the total repair bill is $196.25, at what time was the repair completed?

Translate the sentence into an equation. The number of hours it took to complete the job is unknown, so call it \begin{align*}h\end{align*}.

Write the equation: \begin{align*}65 + 75(h) = 196.25.\end{align*}

Apply the Addition Property of Equality and simplify:

\begin{align*}65 + 75(h)-65 = 196.25-65\end{align*}

\begin{align*}75(h)=131.25\end{align*}

Apply the Multiplication Property of Equality and simplify:

\begin{align*}75(h)\div 75 = 131.25 \div 75\\ h=1.75\end{align*}

The plumber worked for 1.75 hours, or 1 hour, 45 minutes. Since he started at 9:30, the repair was completed at 11:15.

#### Example 3

To determine the temperature in Fahrenheit, multiply the Celsius temperature by 1.8 and then add 32. Determine the Celsius temperature if it is \begin{align*}89^\circ F\end{align*}.

Translate the sentence into an equation. The temperature in Celsius is unknown; call it \begin{align*}C\end{align*}.

Write the equation: \begin{align*}1.8C + 32 = 89.\end{align*}

Apply the Addition Property and simplify.

\begin{align*}1.8C + 32 - 32 & = 89 - 32 \\ 1.8C & = 57\end{align*}

Apply the Multiplication Property of Equality: \begin{align*}1.8C \div 1.8 = 57 \div 1.8.\end{align*}

Simplify: \begin{align*}C = 31.67.\end{align*}

If the temperature is \begin{align*}89^\circ F\end{align*}, then it is \begin{align*}31.67^\circ C\end{align*}.

### Review

- Define like terms. Give an example of a pair of like terms and a pair of unlike terms.
- Define coefficient.

In 3 – 7, combine the like terms.

- \begin{align*}-7x + 39x\end{align*}
- \begin{align*}3x^2 + 21x + 5x + 10x^2\end{align*}
- \begin{align*}6xy + 7y + 5x + 9xy \end{align*}
- \begin{align*}10ab + 9-2ab \end{align*}
- \begin{align*}-7mn-2mn^2-2mn + 8\end{align*}
- Explain the procedure used to solve \begin{align*}-5y-9=74\end{align*}.

Solve and check your solution.

- \begin{align*}1.3x - 0.7x = 12\end{align*}
- \begin{align*}6x-1.3=3.2\end{align*}
- \begin{align*}5x-(3x+2)=1\end{align*}
- \begin{align*}4(x+3)=1\end{align*}
- \begin{align*}5q - 7 = \frac{2}{3}\end{align*}
- \begin{align*}\frac{3}{5}x + \frac{5}{2} = \frac{2}{3}\end{align*}
- \begin{align*}s - \frac{3s}{8} = \frac{5}{6}\end{align*}
- \begin{align*}0.1y + 11 =0\end{align*}
- \begin{align*}\frac{5q-7}{12} = \frac{2}{3}\end{align*}
- \begin{align*}\frac{5(q-7)}{12} = \frac{2}{3}\end{align*}
- \begin{align*}33t - 99 = 0\end{align*}
- \begin{align*}5p - 2 =32\end{align*}
- \begin{align*}14x + 9x = 161\end{align*}
- \begin{align*}3m - 1 + 4m = 5\end{align*}
- \begin{align*}8x + 3 = 11\end{align*}
- \begin{align*}24 = 2x + 6\end{align*}
- \begin{align*}66 = \frac{2}{3}k\end{align*}
- \begin{align*}\frac{5}{8} = \frac{1}{2}(a + 2)\end{align*}
- \begin{align*}16 = -3d - 5\end{align*}
- Jayden purchased a new pair of shoes. Including a 7% sales tax, he paid $84.68. How much did his shoes cost before sales tax?
- A mechanic charges $98 for parts and $60 per hour for labor. Your bill totals $498.00, including parts and labor. How many hours did the mechanic work?
- An electric guitar and amp set costs $1195.00. You are going to pay $250 as a down payment and pay the rest in 5 equal installments. How much should you pay each month?
- Jade is stranded downtown with only $10 to get home. Taxis cost $0.75 per mile, but there is an additional $2.35 hire charge. Write a formula and use it to calculate how many miles she can travel with her money. Determine how many miles she can ride.
- Jasmin’s dad is planning a surprise birthday party for her. He will hire a bouncy castle and provide party food for all the guests. The bouncy castle costs $150 dollars for the afternoon, and the food will cost $3.00 per person. Andrew, Jasmin’s dad, has a budget of $300. Write an equation to help him determine the maximum number of guests he can invite.

**Mixed Review**

- Trish showed her work solving the following equation. What did she do incorrectly? \begin{align*}-2c & = 36 \\ c & = 18\end{align*}
- Write an expression for the following situation: Yoshi had \begin{align*}d\end{align*} dollars, spent $65, and earned $12. He had $96 left.
- Find the domain of the following graph.
- Does the graph represent a function? Explain your answer.
- Find the difference: \begin{align*}\frac{1}{2}- \frac{15}{9}\end{align*}.
- What is the additive identity?
- Find the opposite of –4.1398.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.3.