# How do you find the x- and y-intercepts for the given function. Then graph the function: 2x-3y=6?

Apr 18, 2018

$\text{x-intercept "=3," y-intercept } = - 2$

#### Explanation:

$\text{to find the intercepts, that is where the graph crosses}$
$\text{the x and y axes}$

• " let x = 0, in the equation for y-intercept"

• " let y = 0, in the equation for x-intercept"

$x = 0 \Rightarrow 0 - 3 y = 6 \Rightarrow y = - 2 \leftarrow \textcolor{red}{\text{y-intercept}}$

$y = 0 \Rightarrow 2 x - 0 = 6 \Rightarrow x = 3 \leftarrow \textcolor{red}{\text{x-intercept}}$

$\text{Plot the points } \left(0 , - 2\right) , \left(3 , 0\right)$

$\text{and draw a straight line through them}$
graph{(y-2/3x+2)((x-3)^2+(y-0)^2-0.04)((x-0)^2+(y+2)^2-0.04)=0 [-10, 10, -5, 5]}

Apr 18, 2018

Set $x$ equal to $0$ to find the $y$-intercept.

Then set $y$ equal to $0$ to find the $x$-intercept.

Answers are $\left(0 , - 2\right) \mathmr{and} \left(3 , 0\right)$, respectively.
Plotting instructions are in the Explanation.

#### Explanation:

First, we'll find our intercepts.

The intercept occurs when the function crosses the axis in question, and that means that the value of the opposite axis is equal to 0.

This means we plug in $x = 0$ to find the y-intercept, and $y = 0$ to find the x-intercept.

$y$-intercept:

$2 x - 3 y = 6$

$2 \left(0\right) - 3 y = 6 \Rightarrow - 3 y = 6$

$y = - 2$

$\textcolor{b l u e}{\Rightarrow y \text{-intercept} = \left(0 , - 2\right)}$

$x$-intercept:

$2 x - 3 y = 6$

$2 x - 3 \left(0\right) = 6 \Rightarrow 2 x = 6$

$x = 3$

$\textcolor{p u r p \le}{\Rightarrow x \text{-intercept} = \left(3 , 0\right)}$

Since we know our intercepts, the EASIEST way to plot our line is to plot both intercepts and then draw a line that passes through both points.

The other way to do it is to re-form the equation into slope-intercept form, $y = m x + b$, and then plot that function:

$2 x - 3 y = 6 \Rightarrow \cancel{2 x} - 3 y \textcolor{red}{- 2 x} = 6 \textcolor{red}{- 2 x}$

$- 3 y = - 2 x + 6 \Rightarrow \frac{\cancel{- 3}}{\cancel{\textcolor{red}{- 3}}} y = \frac{- 2}{\textcolor{red}{- 3}} x + \frac{6}{\textcolor{red}{- 3}}$

$y = \frac{2}{3} x - 2$

And now plot like you would plot any other slope-intercept function:

graph{y=2/3x-2 [-10, 10, -5, 5]}

Note that the line passes through both intercepts, so either way works!