# 1.8: Equations that Describe Patterns

**Advanced**Created by: CK-12

**Practice**Equations that Describe Patterns

Suppose that you have 45 minutes to do your math homework, and your teacher has assigned 15 problems. To find out, on average, how many minutes you can spend on each problem, what equation could you set up? Also, if you set up an equation and solve it to find the answer, how will you know that your answer is correct? In this Concept, you will learn how to decide what equation to use and how to make sure your answer is correct once you've found it.

### Guidance

When an algebraic expression is set equal to another value, variable, or expression, a new mathematical sentence is created. This sentence is called an **equation.**

**Definition:** An **algebraic equation** is a mathematical sentence connecting an expression to a value, a variable, or another expression with an equal sign (=).

Suppose there is a concession stand at a theme park selling burgers and French fries. Each burger costs $2.50 and each order of French fries costs $1.75. You and your family will spend exactly $25.00 on food. How many burgers can be purchased? How many orders of fries? How many of each type can be purchased if your family plans to buy a combination of burgers and fries?

The underlined word exactly lends a clue to the type of mathematical sentence you will need to write to model this situation.

These words can be used to symbolize the equal sign:

*Exactly, equivalent, the same as, identical, is*

The word *exactly* is synonymous with equal, so this word is directing us to write an equation. Using the methods learned in lessons 1.1 and 1.6, read every word in the sentence and translate each into mathematical symbols.

#### Example A

Your family is planning to purchase only burgers. How many can be purchased with $25.00?

**Solution:**

Step 1: Choose a variable to represent the unknown quantity, say \begin{align*}b\end{align*} for burgers.

Step 2: Write an equation to represent the situation: \begin{align*}2.50 b = 25.00\end{align*}.

Step 3: Think. What number multiplied by 2.50 equals 25.00?

That number is 10. Checking an answer to an equation is almost as important as the equation itself. By substituting the value for the variable, you are making sure both sides of the equation balance. Let's check that 10 is the solution to our equation by substituting it back in for \begin{align*}b\end{align*}.

\begin{align*}2.50 (10) = 25.00\end{align*}

\begin{align*}25.00 = 25.00\end{align*}

Since these numbers are equal, 10 is the solution. Your family can purchase exactly ten burgers

#### Example B

Is \begin{align*}z = 3\end{align*} a solution to \begin{align*}z^2 + 2z = 8\end{align*}?

**Solution:** Begin by substituting the value of 3 for \begin{align*}z\end{align*}.

\begin{align*}3^2 + 2(3) &= 8\\ 9 + 6 &= 8\\ 15 &= 8\end{align*}

Because \begin{align*}15 = 8\end{align*} is NOT a true statement, we can conclude that \begin{align*}z = 3\end{align*} is not a solution to \begin{align*}z^2 + 2z = 8\end{align*}.

#### Example C

Check that \begin{align*}x = 5\end{align*} is the solution to the equation \begin{align*}3x + 2 = -2x + 27\end{align*}.

**Solution:** To check that \begin{align*}x = 5\end{align*} is the solution to the equation, substitute the value of 5 for the variable, \begin{align*}x\end{align*}:

\begin{align*}3x + 2 &= -2x + 27\\ 3 \cdot x + 2 &= -2 \cdot x + 27\\ 3 \cdot 5 + 2 &= -2 \cdot 5 + 27\\ 15 + 2 &= -10 + 27\\ 17 &= 17\end{align*}

Because \begin{align*}17 = 17\end{align*} is a true statement, we can conclude that \begin{align*}x = 5\end{align*} is a solution to \begin{align*}3x + 2 = -2x + 27\end{align*}.

### Vocabulary

**Algebraic equation:** An ** algebraic equation** is a mathematical sentence connecting an expression to a value, a variable, or another expression with an equal sign (=).

### Guided Practice

Translate the following into equations:

a) *9 less than twice a number is* 33.

b) *Five more than four times a number is* 21.

c) $20.00 *was one-quarter of the money spent on pizza.*

**Solutions:**

a) Let “a number” be \begin{align*}n\end{align*}. So, twice a number is \begin{align*}2n\end{align*}.

Nine less than that is \begin{align*}2n - 9\end{align*}.

The word *is* means the equal sign, so \begin{align*}2n - 9 = 33\end{align*}.

b) Let “a number” be \begin{align*}x\end{align*}. So five more than four times a number is 21 can be written as: \begin{align*}4x + 5 = 21\end{align*}.

c) Let “of the money” be \begin{align*}m\end{align*}. The equation could be written as \begin{align*}\frac{1}{4} m = 20.00\end{align*}.

**More Practice**

For c) above, find how much money was spent on pizza.

\begin{align*}\frac{1}{4} m = 20.00\end{align*}

Think: One-quarter can also be thought of as *divide by four.* What divided by 4 equals 20.00?

The solution is 80. So, the money spent on pizza was $80.00.

### 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: Equations and Inequalities (16:11)

In 1 – 3, define the variables and translate the following statements into algebraic equations.

- Peter’s Lawn Mowing Service charges $10 per job and $0.20 per square yard. Peter earns $25 for a job.
- Renting the ice-skating rink for a birthday party costs $200 plus $4 per person. The rental costs $324 in total.
- Renting a car costs $55 per day plus $0.45 per mile. The cost of the rental is $100.

In 4 – 7, check that the given number is a solution to the corresponding equation.

- \begin{align*}a = -3; \ 4a + 3 = -9\end{align*}
- \begin{align*}x = \frac{4}{3}; \ \frac{3}{4}x + \frac{1}{2} = \frac{3}{2}\end{align*}
- \begin{align*}y = 2; \ 2.5y - 10.0 = -5.0\end{align*}
- \begin{align*}z = -5; \ 2(5 - 2z) = 20 - 2(z - 1)\end{align*}

In 8-12, find the value of the variable.

- \begin{align*}m + 3 = 10\end{align*}
- \begin{align*}6 \times k = 96\end{align*}
- \begin{align*}9 - f = 1\end{align*}
- \begin{align*}8h = 808\end{align*}
- \begin{align*}a + 348 = 0\end{align*}

In 13-15, answer by writing an equation and solving for the variable.

- You are having a party and are making sliders. Each person will eat 5 sliders. There will be seven people at your party. How many sliders do you need to make?
- The cost of a Ford Focus is 27% of the price of a Lexus GS 450h. If the price of the Ford is $15,000, what is the price of the Lexus?
- Suppose your family will purchase only orders of French fries using the information found in the opener of this lesson. How many orders of fries can be purchased for $25.00?

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### Image Attributions

Here you will learn how to read about a real-life situation and write an equation that represents this situation. You will then solve the equation and plug the answer back into the equation to check your work.