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# Intercepts by Substitution

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Practice Intercepts by Substitution
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Using Intercepts

Have you ever tried to choose between two things that you really wanted to do? Take a look at this dilemma.

The seventh graders have to decide between two field trips. One is bowling and the other is a trip to the Omnitheater. It is time for them to figure out which field trip to attend. Mr. Thomas has scheduled a meeting during homeroom to discuss options.

“I think we should go to the Omni Theater because it is more educational,” Tasha stated strongly.

“This trip doesn’t have to be educational we can just have a fun field trip,” Casey said.

Arguments began between the students. Mr. Thomas whistled and all were quiet.

“Is there anything that the two have in common that we can think of?” Mr. Thomas asked.

“You mean money?” Casey asked.

“Yes. Is there a common fee between them both?”

The students began to think about this. Intercepts are places where two or more equations meet. In the case of the students, they wrote two equations one for each field trip. Is there a common cost in each?

Use this Concept to learn about intercepts and then return to this problem and Mr. Thomas’ question at the end of it.

### Guidance

In football, a player makes an interception when he catches a ball thrown by the other team that was not intended for him. Intercept means to catch or to interrupt. In graphs, we will find that most lines intercept the \begin{align*}x\end{align*} - and \begin{align*}y\end{align*} -axes. We’ll call these points the \begin{align*}x\end{align*} - and \begin{align*}y\end{align*} -intercepts and we will use them in a variety of ways.

Consider the graph below. As you know, the \begin{align*}x\end{align*} -axis is horizontal and the \begin{align*}y\end{align*} -axis is vertical. Do you see that the graph crosses, or intercepts, both the \begin{align*}x\end{align*} - and \begin{align*}y\end{align*} -axes?

In looking at this graph, you can see that the line crosses the \begin{align*}x\end{align*} -axis and the \begin{align*}y\end{align*} -axis. These are the two intercepts of this graph.

The line crosses the \begin{align*}x\end{align*} – axis at -1.

The line crosses the \begin{align*}y\end{align*} – axis at 3.

These are the two intercepts.

We can figure out the two intercepts of any linear equation. All you have to do is to look for the place where the line crosses the two axis’.

That is a great question. Because a vertical line is \begin{align*}x =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*} , it would not have a \begin{align*}y\end{align*} – intercept. A horizontal line has the equation \begin{align*}y =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*} , so it does not have an \begin{align*}x\end{align*} – intercept. Therefore, we would only name the \begin{align*}x\end{align*} – intercept or the \begin{align*}y\end{align*} – intercept in these two examples. Here is a graph of both a horizontal line and a vertical line. You can see what we mean by looking at this one.

In these two graphs, \begin{align*}x\end{align*} is equal to 4 and \begin{align*}y\end{align*} is equal to -1. You can see that each of these special types of graphs only has one intercept.

Find the \begin{align*}x\end{align*} - and \begin{align*}y\end{align*} -intercepts and then graph the equation \begin{align*}2x+3y=6\end{align*} .

First, notice that this is an equation in standard form. We will need to find the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} – intercepts.

To find the \begin{align*}x\end{align*} -intercept, set \begin{align*}y\end{align*} equal to zero. Think about this and it makes perfect sense. If you have an intercept with the \begin{align*}x\end{align*} – axis, then it makes sense that the \begin{align*}y\end{align*} value is 0.

\begin{align*}2x+3y&=6\\ 2x+3 \cdot 0&=6\\ 2x&=6\\ x&=3\end{align*}

We now have the ordered pair (3, 0) or the \begin{align*}x\end{align*} -intercept 3.

To find the \begin{align*}y\end{align*} -intercept set \begin{align*}x\end{align*} equal to zero. Think about this and it makes perfect sense. If you have an intercept with the \begin{align*}y\end{align*} – axis, then it makes sense that the \begin{align*}x\end{align*} value is 0.

\begin{align*}2x+3y&=6\\ 2 \cdot 0+3y&=6\\ 3&=6\\ y&=2\end{align*}

We now have the ordered pair (0, 2) or the \begin{align*}y\end{align*} -intercept 2.

Consider a graph whose \begin{align*}x\end{align*} -intercept is 5. Not only does this indicate to us that the graph will cross the \begin{align*}x\end{align*} -axis at 5, but it necessitates that when \begin{align*}x\end{align*} is 5, the \begin{align*}y\end{align*} value is zero. Likewise, for whatever value the \begin{align*}y\end{align*} -intercept has, the \begin{align*}x\end{align*} value must be zero.

Look at the following graph and interpret the intercepts of the graph.

Now let’s look at what information we can interpret from this graph. First, this is a graph of the equation \begin{align*}y=-2x-4\end{align*} .

Notice that the coordinates of the \begin{align*}y\end{align*} – intercept is (0, -4). We can see that the -4 can also be found in the equation itself. Notice how it is the value that is not connected to the \begin{align*}x\end{align*} variable. When looking at an equation and a graph, this is one way to determine the \begin{align*}y\end{align*} – intercept.

Now we can look at the value of the \begin{align*}x\end{align*} -intercept. In this case, it is (-2, 0). Don’t let this fool you, the \begin{align*}y\end{align*} – intercept can be found in the equation, but the \begin{align*}x\end{align*} – intercept is determined by the steepness of the line. Therefore, we will have to use the equation and a table of values to determine the \begin{align*}x\end{align*} – intercept. However, you can still determine it when looking at the graph of a line.

Determine the x and y-intercepts for each equation.

#### Example A

\begin{align*}2x + 4y = 8\end{align*}

Solution: (4,0) and (0,2)

#### Example B

\begin{align*}3x+2y=6\end{align*}

Solution: (2,0)(0,3)

#### Example C

\begin{align*}4x-3y=12\end{align*}

Solution: (3,0)(0,-4)

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

To determine the intercept, we must first begin with the two equations.

The bowling trip used the equation \begin{align*}y = 3g + 2\end{align*} .

The Omni trip used the equation \begin{align*}y=5x+2\end{align*}

You might notice right away that the 2 is common in both. We can check and see if this is indeed the intercept by graphing both equations. Here is the graph.

The \$2.00 fee for shoes or ticket service fee is the common factor between both trips.

### Guided Practice

Here is one for you to try on your own.

Martha likes to go to the park every day but it’s 6 o’clock and her parents are waiting for her at home. She has her bike but she sometimes walks it. She walks at 3mph and she rides her bike at 9mph. If she is 6 miles from home, how long might her parents have to wait?

Solution

\begin{align*}w=\end{align*} time (in hours) walking and \begin{align*}b=\end{align*} time(in hours) on her bike

\begin{align*}3w+9b &=6\\ 3 \cdot 0+9b &=6\\ 9b &=6\\ b &=\frac{2}{3}\end{align*}

If she only rides her bike, it will take her \begin{align*}\frac{2}{3}\end{align*} or 40 minutes.

\begin{align*}3 \cdot w + 9 \cdot 0 &=6\\ 3w &=6\\ w &=2\end{align*}

If she only walks, it will take her 2 hours! Hope she rides her bike.

### Explore More

Directions: Determine the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} – intercepts of each equation. There will be two answers for each equation.

1. \begin{align*}3x+4y=12\end{align*}
2. \begin{align*}6x + 2y = 12\end{align*}
3. \begin{align*}4x + 5y = 20\end{align*}
4. \begin{align*}4x + 2y = 8\end{align*}
5. \begin{align*}3x + 5y = 15\end{align*}
6. \begin{align*}-2x + 3y = -6\end{align*}
7. \begin{align*}-3x + y = 9\end{align*}
8. \begin{align*}-2x - 2y = 6\end{align*}
9. \begin{align*}7x + 3y = 21\end{align*}
10. \begin{align*}2x + 9y = 36\end{align*}

Directions: Look at each graph and identify the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} – intercept of each equation. Each graph will have two answers.

### Vocabulary Language: English

$x-$intercept

$x-$intercept

An $x-$intercept is a location where a graph crosses the $x-$axis. As a coordinate pair, this point will always have the form $(x, 0)$. $x-$intercepts are also called solutions, roots or zeros.
$y-$intercept

$y-$intercept

A $y-$intercept is a location where a graph crosses the $y-$axis. As a coordinate pair, this point will always have the form $(0, y)$.
Intercept

Intercept

The intercepts of a curve are the locations where the curve intersects the $x$ and $y$ axes. An $x$ intercept is a point at which the curve intersects the $x$-axis. A $y$ intercept is a point at which the curve intersects the $y$-axis.
Intercept Method

Intercept Method

The intercept method is a way of graphing a linear function by using the coordinates of the $x$ and $y$-intercepts. The graph is drawn by plotting these coordinates on the Cartesian plane and then joining them with a straight line.

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