# How do you find the x and y intercepts for 9x = 54 - 6y?

Mar 9, 2018

This is in the quadratic question sub group. Did you mean $9 {x}^{2} = 54 - 6 y$?

${y}_{\text{intecpt}} = 9$

$x = - \sqrt{6} \mathmr{and} x = + \sqrt{6}$

#### Explanation:

To make the $y$ term positive multiply everything on both sides by (-1) giving:

$- 9 {x}^{2} = - 54 + 6 y$

Isolate the $y$ term: Add 54 to both sides

$54 - 9 {x}^{2} = 6 y$

Isolate $y$: Divide throughout by 6

$\frac{54}{6} - \frac{9}{6} x = y$

Write as per convention

$y = - \frac{9}{6} {x}^{2} + \frac{54}{6}$

$y = - \frac{3}{2} {x}^{2} + 9$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the y-intercept}}$

This occurs at $x = 0$

${y}_{\text{intecpt}} = - \frac{3}{2} \left({0}^{2}\right) + 9 = 9$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\text{Determine the x-intercept}}$

This occurs at $y = 0$

${x}_{\text{intecpt}} \to y = 0 = - \frac{3}{2} {x}^{2} + 9$

$\frac{3}{2} {x}^{2} = 9$

${x}^{2} = 9 \times \frac{2}{3} = 6$

$x = - \sqrt{6} \mathmr{and} x = + \sqrt{6}$

Mar 9, 2018

Suppose you really meant $9 x = 54 - 6 y$

${y}_{\text{intercept}} = 9$

${x}_{\text{intercept}} = 6$

#### Explanation:

Multiply both sides by (-1)

$- 9 x = - 54 + 6 y$

$- 9 x + 54 = 6 y$

Divide both side by 6

$- \frac{9}{6} x + \frac{54}{6} = y$

$y = - \frac{3}{2} x + 9$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the y-intercept}}$

This occurs at $x = 0$

$y = - \frac{3}{2} \left(0\right) + 9 = 9$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the x-intercept}}$

This occurs at $y = 0$

$y = 0 = - \frac{3}{2} x + 9$

Add $\frac{3}{2} x$ to both sides. Moves it to the left of the = sign.

$+ \frac{3}{2} x = 9$

Multiply both sides by $\frac{2}{3}$

$\frac{3}{2} \times \frac{2}{3} \times x = 9 \times \frac{2}{3}$

Using the principle that $2 \times 3 = 6 = 3 \times 2 \to$ you can move them around.

$\frac{3}{3} \times \frac{2}{2} \times x = 9 \times \frac{2}{3}$

$1 \textcolor{w h i t e}{\text{d}} \times 1 \times x = 9 \times \frac{2}{3}$

${x}_{\text{intercept}} = 6$

color(white)("d")