Evaluate Using the Given Value y(x/3)*3y=x+2

$y (\frac{x}{3}) \cdot 3 y = x + 2$

Step 1

Simplify.

Combine $y$ and $\frac{x}{3}$ .

$\frac{y x}{3} \cdot (3 y) = x + 2$

Combine $3$ and $\frac{y x}{3}$ .

$\frac{3 (y x)}{3} \cdot y = x + 2$

Combine $\frac{3 (y x)}{3}$ and $y$ .

$\frac{3 (y x) y}{3} = x + 2$

Raise $y$ to the power of $1$ .

$\frac{3 (x) (y^{1} y)}{3} = x + 2$

Raise $y$ to the power of $1$ .

$\frac{3 (x) (y^{1} y^{1})}{3} = x + 2$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (x) y^{1 + 1}}{3} = x + 2$

Add $1$ and $1$ .

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

Multiply $3$ by $x$ .

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

Step 2

Multiply each term in $\frac{3 x y^{2}}{3} = x + 2$ by $\frac{3}{x \cdot 3}$ to eliminate the fractions.

Multiply each term in $\frac{3 x y^{2}}{3} = x + 2$ by $\frac{3}{x \cdot 3}$ .

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

Simplify the left side.

Combine.

$\frac{3 x y^{2} \cdot 3}{3 (x \cdot 3)} = x \frac{3}{x \cdot 3} + 2 \frac{3}{x \cdot 3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x y^{2} \cdot 3}{3 (x \cdot 3)} = x \frac{3}{x \cdot 3} + 2 \frac{3}{x \cdot 3}$

Rewrite the expression.

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Simplify each term.

Cancel the common factor of $x$ .

Factor $x$ out of $x \cdot 3$ .

$y^{2} = x \frac{3}{x (3)} + 2 \frac{3}{x \cdot 3}$

Cancel the common factor.

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

Rewrite the expression.

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

Divide $3$ by $3$ .

$y^{2} = 1 + 2 \frac{3}{x \cdot 3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y^{2} = 1 + 2 \frac{3}{x \cdot 3}$

Rewrite the expression.

$y^{2} = 1 + 2 \frac{1}{x}$

Combine $2$ and $\frac{1}{x}$ .

$y^{2} = 1 + \frac{2}{x}$

Step 3

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

$y = \pm \sqrt{1 + \frac{2}{x}}$

Step 4

Simplify $\pm \sqrt{1 + \frac{2}{x}}$ .

Write $1$ as a fraction with a common denominator.

$y = \pm \sqrt{\frac{x}{x} + \frac{2}{x}}$

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x + 2}{x}}$

Rewrite $\sqrt{\frac{x + 2}{x}}$ as $\frac{\sqrt{x + 2}}{\sqrt{x}}$ .

$y = \pm \frac{\sqrt{x + 2}}{\sqrt{x}}$

Multiply $\frac{\sqrt{x + 2}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$y = \pm \frac{\sqrt{x + 2}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{x + 2}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$y = \pm \frac{\sqrt{x + 2} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Raise $\sqrt{x}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x + 2} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}$

Raise $\sqrt{x}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x + 2} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{x + 2} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{x + 2} \sqrt{x}}{{\sqrt{x}}^{2}}$

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{x + 2} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \pm \frac{\sqrt{x + 2} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$y = \pm \frac{\sqrt{x + 2} \sqrt{x}}{x^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \pm \frac{\sqrt{x + 2} \sqrt{x}}{x^{\frac{2}{2}}}$

Rewrite the expression.

$y = \pm \frac{\sqrt{x + 2} \sqrt{x}}{x^{1}}$

Simplify.

$y = \pm \frac{\sqrt{x + 2} \sqrt{x}}{x}$

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(x + 2) x}}{x}$

Reorder factors in $\pm \frac{\sqrt{(x + 2) x}}{x}$ .

$y = \pm \frac{\sqrt{x (x + 2)}}{x}$

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.

$y = \frac{\sqrt{x (x + 2)}}{x}$

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

$y = - \frac{\sqrt{x (x + 2)}}{x}$

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

$y = \frac{\sqrt{x (x + 2)}}{x}$

$y = - \frac{\sqrt{x (x + 2)}}{x}$

$y = \frac{\sqrt{x (x + 2)}}{x}$

$y = - \frac{\sqrt{x (x + 2)}}{x}$

Evaluate Using the Given Value y(x/3)*3y=x+2

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