Suppose that you know that two points on a line are \begin{align*}(2, -4)\end{align*} and \begin{align*}(3, -2)\end{align*}. How would you go about finding the "steepness" of the line? As you'll learn in a future concept, it is the difference of the dependent variable divided by the difference of the independent variable.

**Subtraction of Rational Numbers**

You have learned how to find the opposite of a rational number and them together and now you can use these two ideas to subtract rational numbers as well. Suppose you want to find the difference of 9 and 12. Symbolically, it would be \begin{align*}9 - 12\end{align*}. Begin by placing a dot at nine and move to the left 12 units.

\begin{align*}9 - 12 = -3\end{align*}

The rule is that to subtract a number, add its opposite as shown below:

\begin{align*}3 - 5 = 3 + (-5) = -2 && 9 - 16 = 9 + (-16) = -7\end{align*}

A special case of this rule can be written when trying to subtract a negative number.

The **Opposite-Opposite Property** states that for any real numbers \begin{align*}a\end{align*} and \begin{align*}b, \ a-(-b) = a + b\end{align*}.

#### Let's apply the information from above and simplify the following expressions:

- \begin{align*}-6 - (-13).\end{align*}

Using the Opposite-Opposite Property, the double negative is rewritten as a positive.

\begin{align*}-6 - (-13) = -6 + 13 = 7\end{align*}

- \begin{align*}\frac{5}{6} - \left ( - \frac{1}{18} \right ).\end{align*}

Begin by using the Opposite-Opposite Property.

\begin{align*}\frac{5}{6} + \frac{1}{18}\end{align*}

Next, create a common denominator: \begin{align*}\frac{5 \times 3}{6 \times 3} + \frac{1}{18} = \frac{15}{18} + \frac{1}{18}.\end{align*}

Add the fractions: \begin{align*}\frac{15}{18} + \frac{1}{18} = \frac{16}{18}.\end{align*}

Reduce: \begin{align*}\frac{2 \times 2 \times 2 \times 2}{3 \times 3 \times 2} = \frac{8}{9}.\end{align*}

#### Evaluating Change Using a Variable Expression

You have learned how to graph a function by using an algebraic expression to generate a table of values. Using the table of values you can find the difference of the dependent values relative to the difference of the two independent values. This will tell you what the average change is in the dependent variable compared to the independent variable.

In a previous concept, you wrote an expression to represent the pattern of the total cost to the number of CDs purchased. The table that you were given is repeated below:

\begin{align*}&\text{Number of CDs} && 2 && 4 && 6 && 8 && 10\\ &\text{Cost (\$)} && 24 && 48 && 72 && 96 && 120\end{align*}

To determine the average cost of the CDs, you must find the difference between the dependent values and divide it by the difference in the independent values.

We begin by finding the difference between the cost of two values. For example, the change in cost between 4 CDs and 8 CDs.

\begin{align*}96-48 = 48\end{align*}

Next, we find the difference between the number of CDs.

\begin{align*}&&&8 - 4 = 4\\ &\text{Finally, we divide.}&& \quad \frac{48}{4}=12\end{align*}

This tells us that the cost for each CD was 12 dollars.

### Examples

#### Example 1

Earlier, you were asked about how you would find the steepness of the line that goes through the two points \begin{align*}(2, -4)\end{align*} and \begin{align*}(3, -2)\end{align*}. This steepness is the difference of the dependent variable divided by the difference of the independent variable.

The dependent variable is the first coordinate, the \begin{align*}x\end{align*}-coordinate. The difference in dependent values is: \begin{align*}3-2=1\end{align*}.

The independent variable is the second coordinate, the \begin{align*}y\end{align*}-coordinate. The difference in independent values is: \begin{align*}-2-(-4)=-2+4=2\end{align*}. Remember to use the Opposite-Opposite property as needed.

Now, dividing the difference in dependent values by the difference in independent values, we get that the steepness is \begin{align*}\frac{1}{2}\end{align*}.

#### Example 2

For the equation \begin{align*}y=3x-5\end{align*}, find the difference between the dependent variable from \begin{align*}x=2\end{align*} and \begin{align*}x=4\end{align*}. Then, find the steepness of the line.

First we find the value of the equation, or the value of \begin{align*}y\end{align*} for each \begin{align*}x\end{align*} value:

\begin{align*}y=3x-5=3(2)-5=6-5=1\end{align*}

\begin{align*}y=3x-5=3(4)-5=12-5=7\end{align*}

Now, we calculate the difference in the dependent values:

\begin{align*}7-1=6\end{align*}

Finally, to find the steepness (also called slope in later concepts) we divide by the difference between the independent values:

\begin{align*}4-2=2\end{align*}

\begin{align*}\frac{6}{2}=3\end{align*}

The steepness is 3. Notice that this is the value in front of \begin{align*}x\end{align*} in the equation. We will learn more about that in future concepts.

### Review

In 1–20, subtract the following rational numbers. Be sure that your answer is in the simplest form.

- \begin{align*}9 - 14\end{align*}
- \begin{align*}2 - 7\end{align*}
- \begin{align*}21 - 8\end{align*}
- \begin{align*}8 - (-14)\end{align*}
- \begin{align*}-11 - (-50)\end{align*}
- \begin{align*}\frac{5}{12} - \frac{9}{18}\end{align*}
- \begin{align*}5.4 - 1.01\end{align*}
- \begin{align*}\frac{2}{3} - \frac{1}{4}\end{align*}
- \begin{align*}\frac{3}{4} - \frac{1}{3}\end{align*}
- \begin{align*}\frac{1}{4} - \left (- \frac{2}{3} \right )\end{align*}
- \begin{align*}\frac{15}{11} - \frac{9}{7}\end{align*}
- \begin{align*}\frac{2}{13} - \frac{1}{11}\end{align*}
- \begin{align*}-\frac{7}{8} - \left (- \frac{8}{3} \right )\end{align*}
- \begin{align*}\frac{7}{27} - \frac{9}{39}\end{align*}
- \begin{align*}\frac{6}{11} - \frac{3}{22}\end{align*}
- \begin{align*}-3.1 - 21.49\end{align*}
- \begin{align*}\frac{13}{64} - \frac{7}{40}\end{align*}
- \begin{align*}\frac{11}{70} - \frac{11}{30}\end{align*}
- \begin{align*}-68 - (-22)\end{align*}
- \begin{align*}\frac{1}{3} - \frac{1}{2}\end{align*}
- Determine the steepness of the line between (1, 9) and (5, –14).
- Consider the equation \begin{align*}y = 3x + 2\end{align*}. Determine the steepness in the line between \begin{align*}x = 3\end{align*} and \begin{align*}x = 7.\end{align*}
- Consider the equation \begin{align*}y = \frac{2}{3}x + \frac{1}{2}\end{align*}. Determine the steepness in the line between \begin{align*}x = 1\end{align*} and \begin{align*}x = 2\end{align*}.
- True or false? If the statement is false, explain your reasoning. The difference of two numbers is less than each number.
- True or false? If the statement is false, explain your reasoning. A number minus its opposite is twice the number.
- KMN stock began the day with a price of $4.83 per share. At the closing bell, the price dropped $0.97 per share. What was the closing price of KMN stock?

In 27–32, evaluate the expression. Assume \begin{align*}a=2, \ b= -3,\end{align*} and \begin{align*}c = -1.5.\end{align*}

- \begin{align*}(a-b)+c\end{align*}
- \begin{align*}|b+c|- a\end{align*}
- \begin{align*}a-(b+c)\end{align*}
- \begin{align*}|b|+ |c|+ a\end{align*}
- \begin{align*}7b+4a\end{align*}
- \begin{align*}(c-a)- b\end{align*}

**Mixed Review**

- Graph the following ordered pairs: \begin{align*}\left \{(0,0),(4,4),(7,1),(3,8) \right \}\end{align*}. Is the relation a function?
- Evaluate the following expression when \begin{align*}m= \left (- \frac{2}{3} \right ): \ \frac{2^3+m}{4}\end{align*}.
- Translate the following into an algebraic equation: Ricky has twelve more dollars than Stacy. Stacy has 5 less dollars than Aaron. The total of the friends’ money is $62.
- Simplify \begin{align*}\frac{1}{3} + \frac{7}{5}\end{align*}.
- Simplify \begin{align*}\frac{21}{4} - \frac{2}{3}\end{align*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.5.