4.5: Equations of Lines from Graphs
Write the equation, in standard form, of the following graph:
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Khan Academy Slope and yIntercept Intuition
Guidance
You can determine the equation of a line from a graph by counting. Find the yintercept (\begin{align*}b\end{align*}
The \begin{align*}y\end{align*}
If you cannot determine the yintercept, you can algebraically determine the equation of a line by using the coordinates of two points on the graph. These two points can be used to calculate the slope of the line by counting and then the yintercept can then be determined algebraically.
To write the equation of a line in standard form, the value of the \begin{align*}y\end{align*}
Example A
Determine the equation of the following graph. Write the equation in slopeintercept form.
Solution:
The \begin{align*}y\end{align*}
Example B
Determine the equation, in slopeintercept form of the line shown on the following graph:
Solution:
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 slopeintercept form is \begin{align*}\boxed{y=\frac{5}{6}x+\frac{3}{2}}\end{align*}
Example C
Determine the equation, in standard form, for the line on the following graph:
Solution:
The \begin{align*}y\end{align*}
\begin{align*}yy_1&=m(xx_1) && \text{Use this formula to determine the equation in standard 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&=3x12\\ 5y{\color{red}3x}&=3x{\color{red}3x}12\\ 5y{\color{red}3x}&=12\\ 5y3x{\color{red}+12}&=12{\color{red}+12}\\ 5y3x+12&=0\\ {\color{red}3x}+5y+12&=0 && \text{The coefficient of} \ x \ \text{cannot be a negative value.}\\ 3x5y12&=0\end{align*}
The equation of the line in standard form is \begin{align*}\boxed{3x5y12=0}\end{align*}
Concept Problem Revisited
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*}yy_1&=m(xx_1)\\
y5&=\frac{3}{4}(x2)\\
y5&=\frac{3}{4}x\frac{6}{4}\\
4(y)4(5)&=4\left(\frac{3}{4}\right)x4\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)\\
4y20&=3x6\\
3x+4y20&=3x3x6\\
3x+4y20&=6\\
3x+4y20+6&=6+6\\
3x+4y14&=0\\
3x4y+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{3x4y+14=0}\end{align*}
Vocabulary
 Slope – Intercept Form
 The slopeintercept form is one method for writing the equation of a line. The slopeintercept form is \begin{align*}y = mx + b\end{align*} where \begin{align*}m\end{align*} refers to the slope and \begin{align*}b\end{align*} identifies the \begin{align*}y\end{align*}intercept.
 Standard Form
 The standard form is another method for writing the equation of a line. The standard form is \begin{align*}Ax + By + C = 0\end{align*} where \begin{align*}A\end{align*} is the coefficient of \begin{align*}x\end{align*}, \begin{align*}B\end{align*} is the coefficient of \begin{align*}y\end{align*} and \begin{align*}C\end{align*} is a constant.
Guided Practice
1. Write the equation, in slopeintercept form, of the following graph:
2. Write the equation, in slopeintercept form, of the following graph:
3. Rewrite the equation of the line from #2 in standard form.
Answers:
1. The first step is to determine the coordinates of the \begin{align*}y\end{align*}intercept.

The \begin{align*}y\end{align*}intercept is (0, 3) so \begin{align*}b=3\end{align*}. The second step is to count to determine the value of the slope. The slope is \begin{align*}\frac{4}{7}\end{align*}. The equation of the line in slopeintercept form is \begin{align*}\boxed{y=\frac{4}{7}x3}\end{align*}
2. The \begin{align*}y\end{align*}intercept is not an exact point on the graph. Therefore begin by determining the slope of the line by counting between two points on the line.
 The coordinates of two points on the line are (1, 0) and (6, 4). The slope is is \begin{align*}\frac{4}{5}\end{align*}. The \begin{align*}y\end{align*}intercept of the line must be calculated by using the slope and one of the points on the line.

\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 slopeintercept form is \begin{align*}\boxed{y=\frac{4}{5}x+\frac{4}{5}}\end{align*}
3. 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+5y4&=0\end{align*}
Practice
For each of the following graphs, write the equation in slopeintercept form:
For each of the following graphs, write the equation in slopeintercept form:
For each of the following graphs, write the equation standard form:
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Learning Objectives
Here you'll learn how to find the equation of a line from its graph.
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Date Created:
Dec 19, 2012Last Modified:
Apr 29, 2014Vocabulary
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