Standard Form
Standard form of a linear function is written as \( Ax + By = C \), where the \( x \) and \( y \) terms are together on one side of the equals sign. There are two methods for graphing a line of a function from standard form. The \(x\) and \(y\)-intercepts can be found from the equation and then graphed, or the equation can be converted into slope-intercept (function) form.
Method #1: Graphing from Standard Form by Identifying Intercepts
To find the \(x\) and \(y\)-intercepts for any given equation, we substitute in zero. Why is that? Let's take a look at the graph below:
Method #1: Graphing from Standard Form by Identifying Intercepts
To find the \(x\) and \(y\)-intercepts for any given equation, we substitute in zero. Why is that? Let's take a look at the graph below:
What is the \(y\)-coordinate for each \(x\)-intercept? That's right, it's zero. So when you are trying to find the \(x\)-intercept from an equation, substitute zero in for \(y\).
What is the \(x\)-coordinate for each \(y\)-intercept? Right again, it's zero! When you are trying to find the \(y\)-intercept from an equation, substitute zero in for \(x\).
What is the \(x\)-coordinate for each \(y\)-intercept? Right again, it's zero! When you are trying to find the \(y\)-intercept from an equation, substitute zero in for \(x\).
Time to give it a try!
Example 1: Find the \(x\) and \(y\)-intercepts of the line of the equation \( 2x - 6y = 12 \). Then graph the line.
To identify the \(x\)-intercept of the line of the equation \(2x - 6y = 12 \), substitute zero in for the \(y\)-value and solve for \(x\). Make sure to write the x-intercept as a coordinate!
\begin{align}&\text{Step 1: Substitute 0 in for y.}\ & \ \ \ 2x-6(0)&=12\\&\text{Step 2: Solve for x.}\ & \ \ \ 2x&=12\\& \ & \ \ \ x&=6\\&\text{Step 3: Write as a coordinate.}\ & \ \ \ \text{x-intercept}: &(6, 0)\end{align}
Example 1: Find the \(x\) and \(y\)-intercepts of the line of the equation \( 2x - 6y = 12 \). Then graph the line.
To identify the \(x\)-intercept of the line of the equation \(2x - 6y = 12 \), substitute zero in for the \(y\)-value and solve for \(x\). Make sure to write the x-intercept as a coordinate!
\begin{align}&\text{Step 1: Substitute 0 in for y.}\ & \ \ \ 2x-6(0)&=12\\&\text{Step 2: Solve for x.}\ & \ \ \ 2x&=12\\& \ & \ \ \ x&=6\\&\text{Step 3: Write as a coordinate.}\ & \ \ \ \text{x-intercept}: &(6, 0)\end{align}
To identify the \(y\)-intercept of the line of the equation \( 2x - 6y = 12 \), substitute zero in for the x-value and solve for y. Make sure to write the \(y\)-intercept as a coordinate!
\begin{align}&\text{Step 1: Substitute 0 in for x.}\ & \ \ \ 2(0)-6y&=12\\&\text{Step 2: Solve for y.}\ & \ \ \ -6y&=12\\& \ & \ \ \ y&=-2\\&\text{Step 3: Write as a coordinate.}\ & \ \ \ \text{y-intercept}: &(0, -2)\end{align}
\begin{align}&\text{Step 1: Substitute 0 in for x.}\ & \ \ \ 2(0)-6y&=12\\&\text{Step 2: Solve for y.}\ & \ \ \ -6y&=12\\& \ & \ \ \ y&=-2\\&\text{Step 3: Write as a coordinate.}\ & \ \ \ \text{y-intercept}: &(0, -2)\end{align}
In order to graph the line of the equation, plot both the \(x\)-intercept and \(y\)-intercept and draw the line through the points.
Sometimes graphing by finding the intercepts is not the best approach to graphing from standard form. Let's take a look at the equation: \( 3x - 5y = -10 \). The intercepts of this equation are \( (-\frac{10}{3}, 0) \) and \( (0, 2) \). Since the x-intercept is not an integer, it would be very difficult to accurately graph the line of this equation. So using Method #2 might be a better approach.
Method #2: Graphing from Standard Form by Converting to Slope-Intercept Form
To convert an equation that is in standard form into slope-intercept form, you need to solve the equation for \(y\).
Example 2: Graph the line of the equation \(3x - 5y = -10 \).
First, convert the equation from standard form into function (slope-intercept) form.
\begin{align}3x-5y&=-10\\-3x\ \ \ \ \ \ \ \ \ &\ \ \ \ -3x\ & &\text{Subtract 3x from both sides.}\\\underline{-5y}&=\underline{-3x}-\underline{10}\\-5 & \ \ \ \ \ -5 \ \ -5\ & &\text{Divide each term by -5.}\\y&=\large\frac{3}{5}\normalsize{x+2}\ & &\text{Equation in function form.} \end{align}
Method #2: Graphing from Standard Form by Converting to Slope-Intercept Form
To convert an equation that is in standard form into slope-intercept form, you need to solve the equation for \(y\).
Example 2: Graph the line of the equation \(3x - 5y = -10 \).
First, convert the equation from standard form into function (slope-intercept) form.
\begin{align}3x-5y&=-10\\-3x\ \ \ \ \ \ \ \ \ &\ \ \ \ -3x\ & &\text{Subtract 3x from both sides.}\\\underline{-5y}&=\underline{-3x}-\underline{10}\\-5 & \ \ \ \ \ -5 \ \ -5\ & &\text{Divide each term by -5.}\\y&=\large\frac{3}{5}\normalsize{x+2}\ & &\text{Equation in function form.} \end{align}
Then, graph the line by plotting the y-intercept \( (0, 2) \), and then using the slope ( \(m = \large\frac{3}{5} \)) to plot as additional point.
Quick Check
Graph the following linear equation in standard form: \( 3x - 2y = 6 \).
Quick Check Solutions
Graph the following linear equation in standard form: \( 3x - 2y = 6 \).
Quick Check Solutions