Write as a Function of y 9x^2+7y=6

$9 x^{2} + 7 y = 6$

Step 1

Subtract $7 y$ from both sides of the equation.

$9 x^{2} = 6 - 7 y$

Step 2

Divide each term in $9 x^{2} = 6 - 7 y$ by $9$ and simplify.

Divide each term in $9 x^{2} = 6 - 7 y$ by $9$ .

$\frac{9 x^{2}}{9} = \frac{6}{9} + \frac{- 7 y}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x^{2}}{9} = \frac{6}{9} + \frac{- 7 y}{9}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{6}{9} + \frac{- 7 y}{9}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $6$ and $9$ .

Factor $3$ out of $6$ .

$x^{2} = \frac{3 (2)}{9} + \frac{- 7 y}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

$x^{2} = \frac{3 \cdot 2}{3 \cdot 3} + \frac{- 7 y}{9}$

Cancel the common factor.

$x^{2} = \frac{3 \cdot 2}{3 \cdot 3} + \frac{- 7 y}{9}$

Rewrite the expression.

$x^{2} = \frac{2}{3} + \frac{- 7 y}{9}$

Move the negative in front of the fraction.

$x^{2} = \frac{2}{3} - \frac{7 y}{9}$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{2}{3} - \frac{7 y}{9}}$

Step 4

Simplify $\pm \sqrt{\frac{2}{3} - \frac{7 y}{9}}$ .

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \pm \sqrt{\frac{2}{3} \cdot \frac{3}{3} - \frac{7 y}{9}}$

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2}{3}$ by $\frac{3}{3}$ .

$x = \pm \sqrt{\frac{2 \cdot 3}{3 \cdot 3} - \frac{7 y}{9}}$

Multiply $3$ by $3$ .

$x = \pm \sqrt{\frac{2 \cdot 3}{9} - \frac{7 y}{9}}$

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{2 \cdot 3 - 7 y}{9}}$

Multiply $2$ by $3$ .

$x = \pm \sqrt{\frac{6 - 7 y}{9}}$

Rewrite $\frac{6 - 7 y}{9}$ as ${(\frac{1}{3})}^{2} (6 - 7 y)$ .

Factor the perfect power $1^{2}$ out of $6 - 7 y$ .

$x = \pm \sqrt{\frac{1^{2} (6 - 7 y)}{9}}$

Factor the perfect power $3^{2}$ out of $9$ .

$x = \pm \sqrt{\frac{1^{2} (6 - 7 y)}{3^{2} \cdot 1}}$

Rearrange the fraction $\frac{1^{2} (6 - 7 y)}{3^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{1}{3})}^{2} (6 - 7 y)}$

Pull terms out from under the radical.

$x = \pm \frac{1}{3} \sqrt{6 - 7 y}$

Combine $\frac{1}{3}$ and $\sqrt{6 - 7 y}$ .

$x = \pm \frac{\sqrt{6 - 7 y}}{3}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{6 - 7 y}}{3}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{6 - 7 y}}{3}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{6 - 7 y}}{3}$

$x = - \frac{\sqrt{6 - 7 y}}{3}$

$x = \frac{\sqrt{6 - 7 y}}{3}$

$x = - \frac{\sqrt{6 - 7 y}}{3}$

Step 6

To rewrite as a function of $y$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $y$ is on the other side.

$f (y) = \frac{\sqrt{6 - 7 y}}{3}, - \frac{\sqrt{6 - 7 y}}{3}$

Write as a Function of y 9x^2+7y=6

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