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# 3.3: Solve Equations Involving Inverse Properties of Subtraction and Multiplication

Difficulty Level: At Grade Created by: CK-12
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Practice Two-Step Equations with Subtraction and Multiplication
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Have you ever looked at a homework problem and wondered how to solve it? Look at this situation that Henry faced.

Henry looked at the first problem on his homework page.

\begin{align*}14x - 9 = 19\end{align*}

Even though he'd been paying attention in class, Henry had no idea how to solve this problem.

Do you know how to solve it? This is a two-step equation involving subtraction and multiplication. This Concept will teach you the steps for solving equations like this one.

### Guidance

You are going to learn how to solve two-step equations with subtraction and multiplication in them. Let’s begin.

To solve a two-step equation, we will need to use more than one inverse operation. Let's take a look at how to solve a two-step equation now. When we perform inverse operations to find the value of a variable, we work to get the variable alone on one side of the equals. This is called isolating the variable. It is one strategy for solving equations. You can use isolating the variable whether you are solving one-step or two-step equations.

Solve for \begin{align*}x\end{align*}: \begin{align*}2x - 9 = 17\end{align*}.

Notice that there are two terms on the left side of the equation, \begin{align*}2x\end{align*} and 9. Our first step should be to use inverse operations to get the term that includes a variable, \begin{align*}2x\end{align*}, by itself on one side of the equal (=) sign.

In the equation, 9 is subtracted from \begin{align*}2x\end{align*}. So, we can use the inverse of subtraction—addition. We can subtract 9 from both sides of the equation.

\begin{align*}2x - 9 & = 17\\ 2x(-9 + 9) & = 17 + 9\\ 2x & = 26\end{align*}

Notice how we rewrote the problem above. Since we are adding a positive number, 9, to a number that is being subtracted from \begin{align*}2x\end{align*}, we can represent this as adding 9 to -9 as we did above: (-9 + 9).

The number 9 is the additive inverse, or opposite, of -9.

We can now use inverse operations to get the \begin{align*}x\end{align*} by itself. Since \begin{align*}2x\end{align*} means \begin{align*}2 \cdot x\end{align*}, we can use the inverse of multiplication—division. We can divide both sides of the equation by 2.

\begin{align*}2x & = 26\\ \frac{2x}{2} & = \frac{26}{2}\\ x & = 13\end{align*}

The value of \begin{align*}x\end{align*} is 13.

Let’s review our steps for solving this two-step equation.

Take a few minutes to write these steps in your notebook.

#### Example A

\begin{align*}9x - 5 = 40\end{align*}

Solution:  \begin{align*}x = 5\end{align*}

#### Example B

\begin{align*}9y - 6 = 66\end{align*}

Solution:  \begin{align*}y = 8\end{align*}

#### Example C

\begin{align*}12a - 4 = 44\end{align*}

Solution:  \begin{align*}a = 4\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

Here is the problem that Henry saw on his page.

\begin{align*}14x - 9 = 19\end{align*}

To solve this problem, we can first add nine to both sides of the equation.

\begin{align*}14x - 9 + 9 = 19 + 9\end{align*}

\begin{align*}14x = 28\end{align*}

Now Henry can solve this as a one-step equation by dividing both sides by 14.

\begin{align*}x = 2\end{align*}

This is the answer to this problem.

### Vocabulary

Equation
a mathematical statement with an equal sign where the quantity on one side of the equation is equal to the quantity on the other side.
Variable
a letter used to represent an unknown quantity.
Algebraic Equation
An equation with at least one variable in it.
One-Step Equation
An algebraic equation with one operation in it.
Two-Step Equation
An algebraic equation with two operations in it.

### Guided Practice

Here is one for you to try on your own.

Eight times a number minus four is equal to ninety - two.

Write a two-step equation and solve for the missing variable.

Solution

First, walk through the words to write the equation.

\begin{align*}8x - 4 = 92\end{align*}

Now solve the for the variable. First, add four to both sides of the equation.

\begin{align*}8x - 4 + 4 = 92 + 4\end{align*}

\begin{align*}8x = 96\end{align*}

Now divide both sides by 8.

\begin{align*}x = 12\end{align*}

### Practice

Directions: Solve each two-step equation that has multiplication and subtraction in it.

1. \begin{align*}4x - 3 = 13\end{align*}
2. \begin{align*}5y - 8 = 22\end{align*}
3. \begin{align*}7x - 11 = 31\end{align*}
4. \begin{align*}8y - 15 = 25\end{align*}
5. \begin{align*}9x - 12 = 42\end{align*}
6. \begin{align*}12y - 9 = 99\end{align*}
7. \begin{align*}2y - 3 = 23\end{align*}
8. \begin{align*}3x - 8 = 19\end{align*}
9. \begin{align*}5y - 2 = 28\end{align*}
10. \begin{align*}7x - 11 = 38\end{align*}
11. \begin{align*}5y - 9 = 51\end{align*}
12. \begin{align*}6a - 12 = 30\end{align*}
13. \begin{align*}9x - 14 = 13\end{align*}
14. \begin{align*}12x - 23 = 49\end{align*}
15. \begin{align*}13y - 3 = 23\end{align*}
16. \begin{align*}18x - 12 = 42\end{align*}

### Vocabulary Language: English

Algebraic Equation

An algebraic equation contains numbers, variables, operations, and an equals sign.

Equation

An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.

One-Step Equation

A one-step equation is an algebraic equation with one operation in it that requires one step to solve.

Two-Step Equation

A two-step equation is an algebraic equation with two operations in it that requires two steps to solve.

Variable

A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.

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