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# Intercepts by Substitution

## Graphing equations that form lines by finding x and y intercepts

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Graphs of Linear Functions from Intercepts

What are the intercepts of \begin{align*}4x+2y=8\end{align*}? How could you use the intercepts to quickly graph the function?

### Guidance

To graph a linear function, you need to plot only two points. These points can then be lined up with a straight edge and joined to graph the straight line. While any two points can be used to graph a linear function, two points in particular that can be used are the \begin{align*}x\end{align*}-intercept and the \begin{align*}y\end{align*}-intercept. Graphing a linear function by plotting the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*} intercepts is often referred to as the intercept method.

The \begin{align*}x\end{align*}-intercept is where the graph crosses the \begin{align*}x\end{align*}-axis. Its coordinates are \begin{align*}(x, 0)\end{align*}. Because all \begin{align*}x\end{align*}-intercepts have a \begin{align*}y\end{align*}-coordinate equal to 0, you can find an \begin{align*}x\end{align*}-intercept by substituting 0 for \begin{align*}y\end{align*} in the equation and solving for \begin{align*}x\end{align*}.

The \begin{align*}y\end{align*}-intercept is where the graph crosses the \begin{align*}y\end{align*}-axis. Its coordinates are \begin{align*}(0, y)\end{align*}. Because all \begin{align*}y\end{align*}-intercepts have a \begin{align*}x\end{align*}-coordinate equal to 0, you can find an \begin{align*}y\end{align*}-intercept by substituting 0 for \begin{align*}x\end{align*} in the equation and solving for \begin{align*}y\end{align*}.

#### Example A

Identify the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercepts for each line.

(a) \begin{align*}2x+y-6=0\end{align*}

(b) \begin{align*}\frac{1}{2}x-4y=4\end{align*}

Solution:

(a)

(b)

#### Example B

Use the intercept method to graph \begin{align*}2x-3y=-12\end{align*}.

Solution:

#### Example C

Use the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercepts of the graph to identify the linear function that matches the graph.

a) \begin{align*}y=2x-8\end{align*}

b) \begin{align*}x-2y+8=0\end{align*}

c) \begin{align*}2x+y-8=0\end{align*}

The \begin{align*}x\end{align*}-intercept is (–8, 0) and the \begin{align*}y\end{align*}-intercept is (0, 4).

Solution: Find the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} intercepts for each equation and see which matches the graph.

a) \begin{align*}x\end{align*} intercept: \begin{align*}0=2x-8 \rightarrow x=4\end{align*}

\begin{align*}y\end{align*} intercept: \begin{align*}y=2(0)-8 \rightarrow y=-8\end{align*}

b) \begin{align*}x\end{align*} intercept: \begin{align*}x-2(0)+8=0 \rightarrow x=-8\end{align*}

\begin{align*}y\end{align*} intercept: \begin{align*}0-2y+8=0 \rightarrow y=4\end{align*}

c) \begin{align*}x\end{align*} intercept: \begin{align*}2x+0-8=0 \rightarrow x=4\end{align*}

\begin{align*}y\end{align*} intercept: \begin{align*}2(0)+y-8=0 \rightarrow y=8\end{align*}

The \begin{align*}x\end{align*} and \begin{align*}y\end{align*} intercepts match for \begin{align*}x-2y+8=0\end{align*} so this is the equation of the line.

#### Concept Problem Revisited

The linear function \begin{align*}4x+2y=8\end{align*} can be graphed by using the intercept method.

Plot the \begin{align*}x\end{align*}-intercept on the \begin{align*}x\end{align*}-axis and the \begin{align*}y\end{align*}-intercept on the \begin{align*}y\end{align*}-axis. Join the two points with a straight line.

### Vocabulary

Intercept Method
The intercept method is a way of graphing a linear function by using the coordinates of the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercepts. The graph is drawn by plotting these coordinates on the Cartesian plane and joining them with a straight line.
\begin{align*}x\end{align*}-intercept
An \begin{align*}x\end{align*}-intercept of a relation is the \begin{align*}x-\end{align*}coordinate of the point where the relation intersects the \begin{align*}x\end{align*}-axis.
\begin{align*}y\end{align*}-intercept
A \begin{align*}y\end{align*}-intercept of a relation is the \begin{align*}y-\end{align*}coordinate of the point where the relation intersects the \begin{align*}y\end{align*}-axis.

### Guided Practice

1. Identify the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercepts of the following linear functions:

(i) \begin{align*}2(x-3)=y+4\end{align*}
(ii) \begin{align*}3x+\frac{2}{3}y-3=0\end{align*}

2. Use the intercept method to graph the following relation:

(i) \begin{align*}5x+2y=-10\end{align*}

3. Use the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercepts of the graph, to match the graph to its function.

(i) \begin{align*}2x+y=6\end{align*}
(ii) \begin{align*}4x-3y-12=0\end{align*}
(iii) \begin{align*}5x+3y=15\end{align*}

Answers:

1. (i)

If you prefer to have both variables on the same side of the equation, this form may also be used. The choice is your preference.

(ii)

2.

3. Identify the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercepts from the graph.

The \begin{align*}x\end{align*}-intercept is (3, 0)

The \begin{align*}y\end{align*}-intercept is (0, -4)

Determine the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercept for each of the functions. If the intercepts match those of the graph, then the linear function will be the one that matches the graph.

(i)

\begin{align*}2x+y=6 \end{align*} is not the linear function for the graph.

(ii)

\begin{align*}4x-3y-12=0\end{align*} is the linear function for the graph.

(iii)

\begin{align*}5x+3y=15\end{align*} is not the linear function for the graph.

### Practice

For 1-10, complete the following table:

Function \begin{align*}x\end{align*}-intercept \begin{align*}y\end{align*}-intercept
\begin{align*}7x-3y=21\end{align*} 1. 2.
\begin{align*}8x-3y+24=0\end{align*} 3. 4.
\begin{align*}\frac{x}{4}-\frac{y}{2}=3\end{align*} 5. 6.
\begin{align*}7x+2y-14=0\end{align*} 7. 8.
\begin{align*}\frac{2}{3}x-\frac{1}{4}y=-2\end{align*} 9. 10.

Use the intercept method to graph each of the linear functions in the above table.

1. \begin{align*}7x-3y=21\end{align*}
2. \begin{align*}8x-3y+24=0\end{align*}
3. \begin{align*}\frac{x}{4}-\frac{y}{2}=3\end{align*}
4. \begin{align*}7x+2y-14=0\end{align*}
5. \begin{align*}\frac{2}{3}x-\frac{1}{4}y=-2\end{align*}

Use the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercepts to match each graph to its function.

a. \begin{align*}7x+5y-35=0\end{align*}
b. \begin{align*}y=5x+10\end{align*}
c. \begin{align*}2x+4y+8=0\end{align*}
d. \begin{align*}2x+y=2\end{align*}
1. .

1. .

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### 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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