Suppose you had a cat, and you gave him 1 treat per day. What if you wanted to graph the number of treats left in his treat jar as a function of the number of days that have passed? You could do it if you knew the intercepts, but how would you find them?

### Intercepts and the Cover-Up Method

Recall that to find the \begin{align*}x-\end{align*}**intercept**, substitute zero for the \begin{align*}y\end{align*}-**value** and that to find the \begin{align*}y-\end{align*}**intercept**, substitute zero for the \begin{align*}x\end{align*}-**value**. This method works for any form of linear equations.

A second method of finding the intercepts is called the Cover-Up Method. Using the Multiplication Property of Zero \begin{align*}a(0)=0\end{align*}, you can “cover-up” the other variable and solve for the intercept you wish to find. This method only works for linear equations that are in the form \begin{align*}ax+by=c.\end{align*}

The cover-up method simply removes one variable at a time, in order to solve for the other variable. The solutions are then the intercepts, since removing a variable, or "covering up" a variable, is the same thing as setting it equal to zero.

#### Let's use the Cover-Up Method to complete the following problems:

- Graph \begin{align*}-7x-3y=21\end{align*}.

To solve for the \begin{align*}y-\end{align*}intercept we set \begin{align*}x=0\end{align*} and cover up the \begin{align*}x\end{align*} term:

\begin{align*}-3y& =21\\ y& =-7 && (0,-7) \ \text{is the} \ y-intercept \text{.}\end{align*}

To solve for the \begin{align*}x-\end{align*}intercept, cover up the \begin{align*}y-\end{align*}variable and solve for \begin{align*}x\end{align*}:

\begin{align*}-7x& =21\\ x& =-3 && (-3,0) \ \text{is the} \ x-intercept \text{.}\end{align*}

Now graph by first plotting the intercepts and then drawing a line through these points.

- Jose has $30 to spend on food for a class barbecue. Hot dogs cost $0.75 each (including the bun) and burgers cost $1.25 (including bun and salad). Plot a graph that shows all the combinations of hot dogs and burgers he could buy for the barbecue, spending exactly $30.

Begin by translating this sentence into an algebraic equation. Let \begin{align*}y=\end{align*} *the number of hot dogs* and \begin{align*}x=\end{align*} *the number of burgers*.

\begin{align*}1.25(x)+ 0.75(y)=30\end{align*}

Find the intercepts of the graph. This example will use the Cover-Up Method. Feel free to use substitution if you prefer.

\begin{align*}0.75y& =30\\ y& =40 && y-intercept(0,40)\end{align*}

\begin{align*}1.25x& =30\\ x& =24 && x-intercept(24,0)\end{align*}

By graphing Jose’s situation, you can determine the combinations of hot dogs and burgers he can purchase for exactly $30.00.

- Graph \begin{align*}y+\frac{3}{2}x=-3\end{align*} using the Cover-Up Method.

First start by "covering up" \begin{align*}y\end{align*}. This results in:

\begin{align*}\text{Start with the equation.} && y+\frac{3}{2}x&=-3\\ \text{Then``cover-up" y.} && \frac{3}{2}x&=-3\\ \text{Solve for x.} && x&=-3 \cdot \frac{2}{3}= -2 \end{align*}

When \begin{align*}y=0\end{align*}, \begin{align*}x=-2\end{align*}. This is the \begin{align*}x\end{align*}-intercept.

Now, "cover-up" \begin{align*}x\end{align*}:

\begin{align*}\text{Start with the equation.} && y+\frac{3}{2}x&=-3\\ \text{Then ``cover-up" x.} && y&=-3\\ \end{align*}

When \begin{align*}x=0\end{align*}, \begin{align*}y=-3\end{align*}. This is the \begin{align*}y\end{align*}-intercept.

Now graph the equation by plotting the two intercepts and connecting them with a line.

### Examples

#### Example 1

Earlier, you were told to suppose that you had a cat that you gave 1 treat to per day and you wanted to graph the number of treats left in his treat jar as a function of the number of days that have passed. How would you find the intercepts?

You could use substitution or the Cover-Up Method. To use the Cover-Up Method, first put the equation into \begin{align*}ax+by=c\end{align*} form. Then, "cover up" \begin{align*}y\end{align*} to find the \begin{align*}x\end{align*}-intercept. Finally, "cover up" \begin{align*}x\end{align*} to find the \begin{align*}y\end{align*}-intercept. With those, you can easily graph the equation.

#### Example 2

Graph \begin{align*}4y-3x=12\end{align*} using the cover-up method.

First start by "covering up" \begin{align*}y\end{align*}. This results in:

\begin{align*}\text{Start with the equation.} && 4y-3x&=12\\ \text{Then ``cover-up" y.} &&-3x&=12\\ \text{Solve for x.} && x&=-4 \end{align*}

When \begin{align*}y=0\end{align*}, \begin{align*}x=-4\end{align*}. This is the \begin{align*}x\end{align*}-intercept.

Now, "cover-up" \begin{align*}x\end{align*}:

\begin{align*}\text{Start with the equation.} && 4y-3x&=12\\ \text{Then ``cover-up" x.} && 4y&=12\\ \text{Solve for y.} && y&=3 \end{align*}

When \begin{align*}x=0\end{align*}, \begin{align*}y=3\end{align*}. This is the \begin{align*}y\end{align*}-intercept.

Now graph the equation by plotting the two intercepts and connecting them with a line.

### Review

- Explain the process of the Cover-Up Method.
- Do you prefer the Substitution Method or the Cover-Up Method? Why?

Find the intercepts of the following equations using the Cover-Up Method.

- \begin{align*}5x-6y=15\end{align*}
- \begin{align*}3x-4y=-5\end{align*}
- \begin{align*}2x+7y=-11\end{align*}
- \begin{align*}5x+10y=25\end{align*}

In 7 – 18, use any method to find the intercepts and then graph the equation.

- \begin{align*}y=2x+3\end{align*}
- \begin{align*}6(x-1)=2(y+3)\end{align*}
- \begin{align*}x-y=5\end{align*}
- \begin{align*}x+y=8\end{align*}
- \begin{align*}4x+9y=0\end{align*}
- \begin{align*}\frac{1}{2} x+4y=12\end{align*}
- \begin{align*}x-2y=4\end{align*}
- \begin{align*}7x-5y=10\end{align*}
- \begin{align*}4x-y=-3\end{align*}
- \begin{align*}x-y=0\end{align*}
- \begin{align*}5x+y=5\end{align*}
- \begin{align*}7x-2y=-6\end{align*}
- What needs to be done to the following equation before you can use either method to find its intercepts? \begin{align*}3(x+2)=2(y+3)\end{align*}
- At the local grocery store, strawberries cost $3.00 per pound and bananas cost $1.00 per pound. If I have $10 to spend between strawberries and bananas, draw a graph to show what combinations of each I can buy and spend exactly $10.
- A movie theater charges $7.50 for adult tickets and $4.50 for children. If the $900 theater takes in ticket sales for a particular screening, draw a graph that depicts the possibilities for the number of adult tickets and the number of child tickets sold.
- In football, touchdowns are worth 6 points, field goals are worth 3 points, and safeties are worth 2 points. Suppose there were no safeties and the team scored 36 points. Graph the situation to determine the combinations of field goals and touchdowns the team could have had.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.6.