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# 4.6: Intercepts and the Cover-Up Method

Difficulty Level: Basic Created by: CK-12
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Practice Intercepts by Substitution

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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? One way would be to use the Cover-Up Method. In this Concept, you'll learn all about the Cover-Up method so that you can create graphs such as this.

### Guidance

In the last Concept, you saw how to find intercepts by substituting the value of zero in for one of the variables.

To find the x\begin{align*}x-\end{align*}intercept, substitute zero for the y\begin{align*}y\end{align*}-value.

To find the y\begin{align*}y-\end{align*}intercept, substitute zero for the x\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 a(0)=0\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 ax+by=c.\begin{align*}ax+by=c.\end{align*} We will see how to use the Cover-Up Method on equations in this form below.

#### Example A

Graph 7x3y=21\begin{align*}-7x-3y=21\end{align*} using its intercepts.

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

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

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

7xx=21=3(3,0) is the xintercept.\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.

#### Example B

Jose has $30 to spend on food for a class barbeque. 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.

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

1.25(x)+0.75(y)=30\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.

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

1.25xx=30=24xintercept(24,0)\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. Understanding the Cover-Up Method 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. Try out this concept in the example below: #### Example C Graph y+32x=3\begin{align*}y+\frac{3}{2}x=-3\end{align*} using the Cover-Up Method. Solution: First start by "covering up" y\begin{align*}y\end{align*}. This results in: Start with the equation.Thencover-up" y.Solve for x.y+32x32xx=3=3=323=2\begin{align*}\text{Start with the equation.} && y+\frac{3}{2}x&=-3\\ \text{Thencover-up" y.} && \frac{3}{2}x&=-3\\ \text{Solve for x.} && x&=-3 \cdot \frac{2}{3}= -2 \end{align*} When y=0\begin{align*}y=0\end{align*}, x=2\begin{align*}x=-2\end{align*}. This is the x\begin{align*}x\end{align*}-intercept. Now, "cover-up" x\begin{align*}x\end{align*}: Start with the equation.Then cover-up" x.y+32xy=3=3\begin{align*}\text{Start with the equation.} && y+\frac{3}{2}x&=-3\\ \text{Then cover-up" x.} && y&=-3\\ \end{align*} When x=0\begin{align*}x=0\end{align*}, y=3\begin{align*}y=-3\end{align*}. This is the y\begin{align*}y\end{align*}-intercept. Now graph the equation by plotting the two intercepts and connecting them with a line. ### Guided Practice Graph 4y3x=12\begin{align*}4y-3x=12\end{align*} using the cover-up method. Solution: First start by "covering up" y\begin{align*}y\end{align*}. This results in: Start with the equation.Then cover-up" y.Solve for x.4y3x3xx=12=12=4\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 y=0\begin{align*}y=0\end{align*}, x=4\begin{align*}x=-4\end{align*}. This is the x\begin{align*}x\end{align*}-intercept. Now, "cover-up" x\begin{align*}x\end{align*}: Start with the equation.Then cover-up" x.Solve for y.4y3x4yy=12=12=3\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 x=0\begin{align*}x=0\end{align*}, y=3\begin{align*}y=3\end{align*}. This is the y\begin{align*}y\end{align*}-intercept. Now graph the equation by plotting the two intercepts and connecting them with a line. ### 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: Graphing Using Intercepts (12:18) 1. Explain the process of the Cover-Up Method. 2. Do you prefer the Substitution Method or the Cover-Up Method? Why? Find the intercepts of the following equations using the Cover-Up Method. 1. 5x6y=15\begin{align*}5x-6y=15\end{align*} 2. 3x4y=5\begin{align*}3x-4y=-5\end{align*} 3. 2x+7y=11\begin{align*}2x+7y=-11\end{align*} 4. 5x+10y=25\begin{align*}5x+10y=25\end{align*} In 7 – 18, use any method to find the intercepts and then graph the equation. 1. y=2x+3\begin{align*}y=2x+3\end{align*} 2. 6(x1)=2(y+3)\begin{align*}6(x-1)=2(y+3)\end{align*} 3. xy=5\begin{align*}x-y=5\end{align*} 4. x+y=8\begin{align*}x+y=8\end{align*} 5. \begin{align*}4x+9y=0\end{align*} 6. \begin{align*}\frac{1}{2} x+4y=12\end{align*} 7. \begin{align*}x-2y=4\end{align*} 8. \begin{align*}7x-5y=10\end{align*} 9. \begin{align*}4x-y=-3\end{align*} 10. \begin{align*}x-y=0\end{align*} 11. \begin{align*}5x+y=5\end{align*} 12. \begin{align*}7x-2y=-6\end{align*} 13. 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*} 14. At the local grocery store, strawberries cost3.00 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. 15. 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.
16. 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.

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TermDefinition
intercept The point at which a graphed equation crosses an axis. The $x-$intercept is an ordered pair at which the line crosses the $x-$axis (the horizontal axis). Its ordered pair has the form $(x,0)$. The $y-$intercept is an ordered pair at which the line crosses the $y-$axis (the vertical axis). Its ordered pair has the form $(0,y)$

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