How do you find the slope and intercept of 2x+3y=6?

Jul 19, 2018

Slope: $- \frac{2}{3}$
x-intercept: $\left(3 , 0\right)$
y-intercept: $\left(0 , 2\right)$

Explanation:

$2 x + 3 y = 6$

To find the slope, first make the equation into slope-intercept form: Subtract $\textcolor{b l u e}{2 x}$ from both sides of the equation:
$2 x + 3 y \quad \textcolor{b l u e}{- \quad 2 x} = 6 \quad \textcolor{b l u e}{- \quad 2 x}$

$3 y = 6 - 2 x$

Divide both sides by $\textcolor{b l u e}{3}$:
$\frac{3 y}{\textcolor{b l u e}{3}} = \frac{6 - 2 x}{\textcolor{b l u e}{3}}$

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

We know that the slope is the value multiplied by $x$, meaning that the slope is $- \frac{2}{3}$.

$- - - - - - - - - - - - - - - - - - - -$

To find the x-intercept, plug in $0$ for $y$ and solve for $x$:
$0 = 2 - \frac{2}{3} x$

Simplify:
$- 2 = - \frac{2}{3} x$

$3 = x$

So the $x$-intercept is at $\left(3 , 0\right)$.

$- - - - - - - - - - - - - - - - - - - - -$

To find the y-intercept, plug in $0$ for $x$ and solve for $y$:
$y = 2 - \frac{2}{3} \left(0\right)$

$y = 2 - 0$

$y = 2$

So the $y$-intercept is at $\left(0 , 2\right)$.

Hope this helps!

Jul 19, 2018

See below:

Explanation:

We can find the $x$-intercept by setting $y$ equal to zero. We get

$2 x = 6 \implies x = 3$

This is our $x$-intercept.

Similarly, we can set $x$ equal to zero to find the $y$-intercept. We get

$3 y = 6 \implies y = 2$

This is our $y$-intercept.

Next, we can convert this equation into slope-intercept form

$y = m x + b$, with slope $m$

We can start by subtracting $2 x$ from both sides to get

$3 y = - 2 x + 6$

Next, divide both sides by $3$ to get

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

We see that our slope, the coefficient on $x$ is $- \frac{2}{3}$.

Hope this helps!