Write the equation, in standard form, of the following graph:

### Equations of Lines from Graphs

You can determine the equation of a line from a graph by counting and performing small calculations. One strategy is to find the y-intercept (\begin{align*}b\end{align*}**slope-intercept form**: \begin{align*}y=mx+b\end{align*}

For example, the \begin{align*}y\end{align*}

If you cannot determine the y-intercept because it does not lie on a precise point, you can algebraically determine the equation of the line by using the coordinates of two point on the graph. You can use these two points to calculate the slope of the line either by counting or algebraically. Then, you can plug one of the points into the slope-intercept equation and solve for the y-intercept \begin{align*}(b)\end{align*}

To write the equation of a line in standard form, the value of the \begin{align*}y\end{align*}

#### Let's find the equation of the line for each of the following graphs:

The \begin{align*}y\end{align*}

The \begin{align*}y\end{align*}

\begin{align*}y&=mx+b\\
-1&=\left(\frac{-5}{6}\right)(3)+b\\
{\color{red}-1}&=\left(\frac{{\color{red}-5}}{{\color{red}\cancel{6}_2}}\right)({\color{red}\cancel{3}})+b\\
-1&=\frac{-5}{2}+b\\
-1{\color{red}+\frac{5}{2}}&=\frac{-5}{2}{\color{red}+\frac{5}{2}}+b\\
-1+\frac{5}{2}&=b\\
{\color{red}\frac{-2}{2}}+\frac{5}{2}&=b\\
\frac{3}{2}&=b\end{align*}

The equation in slope-intercept form is \begin{align*}\boxed{y=-\frac{5}{6}x+\frac{3}{2}}\end{align*}

The \begin{align*}y\end{align*}

\begin{align*}y-y_1&=m(x-x_1) && \text{Start with point-slope form.}\\
y-{\color{red}0}&={\color{red}\frac{3}{5}}(x-{\color{red}4}) && \text{Fill in the value for} \ m \ \text{of} \ \frac{3}{5} \ \text{and} \ \begin{pmatrix} x_1, & y_1 \\ 4, & 0 \end{pmatrix}\\
y&=\frac{3}{5}{\color{red}x}-{\color{red}\frac{12}{5}}\\
{\color{red}5}(y)&={\color{red}5}\left(\frac{3}{5}x\right)-{\color{red}5}\left(\frac{12}{5}\right) && \text{Multiply every term by 5.}\\
{\color{red}5}(y)&={\color{red}\cancel{5}}\left(\frac{3}{\cancel{5}}x\right)-{\color{red}\cancel{5}}\left(\frac{12}{\cancel{5}}\right) && \text{Simplify and set the equation equal to zero.}\\
\\
5y&=3x-12\\
5y{\color{red}-3x}&=3x{\color{red}-3x}-12\\
5y{\color{red}-3x}&=-12\\
5y-3x{\color{red}+12}&=-12{\color{red}+12}\\
5y-3x+12&=0\\
{\color{red}-3x}+5y+12&=0 && \text{The coefficient of} \ x \ \text{cannot be a negative value.}\\
3x-5y-12&=0\end{align*}

The equation of the line in standard form is \begin{align*}\boxed{3x-5y-12=0}\end{align*}

### Examples

#### Example 1

Earlier, you were asked to write the equation, in standard form, of the following graph:

The first step is to determine the slope of the line.

- The slope of the line is \begin{align*}\frac{3}{4}\end{align*}
34 . The coordinates of one point on the line are (2, 5). - \begin{align*}y-y_1&=m(x-x_1)\\
y-5&=\frac{3}{4}(x-2)\\
y-5&=\frac{3}{4}x-\frac{6}{4}\\
4(y)-4(5)&=4\left(\frac{3}{4}\right)x-4\left(\frac{6}{4}\right)\\
4(y)-4(5)&=\cancel{4}\left(\frac{3}{\cancel{4}}\right)x-\cancel{4}\left(\frac{6}{\cancel{4}}\right)\\
4y-20&=3x-6\\
-3x+4y-20&=3x-3x-6\\
-3x+4y-20&=-6\\
-3x+4y-20+6&=-6+6\\
-3x+4y-14&=0\\
3x-4y+14&=0\end{align*}
y−y1y−5y−54(y)−4(5)4(y)−4(5)4y−20−3x+4y−20−3x+4y−20−3x+4y−20+6−3x+4y−143x−4y+14=m(x−x1)=34(x−2)=34x−64=4(34)x−4(64)=4(34)x−4(64)=3x−6=3x−3x−6=−6=−6+6=0=0

The equation of the line in standard form is \begin{align*}\boxed{3x-4y+14=0}\end{align*}

#### Example 2

Write the equation, in slope-intercept form, of the following graph:

The first step is to determine the coordinates of the \begin{align*}y\end{align*}

#### Example 3

Write the equation, in slope-intercept form, of the following graph:

The \begin{align*}y\end{align*}

- \begin{align*}y&=mx+b\\ {\color{red}0}&={\color{red}\frac{-4}{5}}({\color{red}1})+b\\ 0&=\frac{-4}{5}+b\\ 0{\color{red}+\frac{4}{5}}&=\frac{-4}{5}{\color{red}+\frac{4}{5}}+b\\ \frac{4}{5}&=b\end{align*}

The equation of the line in slope-intercept form is \begin{align*}\boxed{y=-\frac{4}{5}x+\frac{4}{5}}\end{align*}

#### Example 4

Rewrite the equation of the line from Example 3 in standard form.

To rewrite the equation in standard form, first multiply the equation by 5 to get rid of the fractions. Then, set the equation equal to 0.

- \begin{align*}y&=-\frac{4}{5}x+\frac{4}{5}\\ 5y&=-4x+4\\ 4x+5y-4&=0\end{align*}

### Review

For each of the following graphs, write the equation in slope-intercept form:

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For each of the following graphs, write the equation in slope-intercept form:

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For each of the following graphs, write the equation standard form:

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- Can you always find the equation of a line from its graph?
- How do you find the equation of a vertical line? What about a horizontal line?
- Rewrite the equation \begin{align*}y=\frac{1}{4}x-5\end{align*} in standard form.
- Rewrite the equation \begin{align*}y=\frac{2}{3}x+1\end{align*} in standard form.
- Rewrite the equation \begin{align*}y=\frac{1}{3}x-\frac{3}{7}\end{align*} in standard form.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.5.