Solve the Rational Equation for y square root of 5y+1 = square root of 3y+1

$\sqrt{5 y + 1} = \sqrt{3 y + 1}$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{5 y + 1}}^{2} = {\sqrt{3 y + 1}}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 y + 1}$ as ${(5 y + 1)}^{\frac{1}{2}}$ .

${({(5 y + 1)}^{\frac{1}{2}})}^{2} = {\sqrt{3 y + 1}}^{2}$

Simplify the left side.

Simplify ${({(5 y + 1)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(5 y + 1)}^{\frac{1}{2}})}^{2}$ .

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

${(5 y + 1)}^{\frac{1}{2} \cdot 2} = {\sqrt{3 y + 1}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(5 y + 1)}^{\frac{1}{2} \cdot 2} = {\sqrt{3 y + 1}}^{2}$

Rewrite the expression.

${(5 y + 1)}^{1} = {\sqrt{3 y + 1}}^{2}$

Simplify.

$5 y + 1 = {\sqrt{3 y + 1}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{3 y + 1}}^{2}$ as $3 y + 1$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 y + 1}$ as ${(3 y + 1)}^{\frac{1}{2}}$ .

$5 y + 1 = {({(3 y + 1)}^{\frac{1}{2}})}^{2}$

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

$5 y + 1 = {(3 y + 1)}^{\frac{1}{2} \cdot 2}$

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

$5 y + 1 = {(3 y + 1)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$5 y + 1 = {(3 y + 1)}^{\frac{2}{2}}$

Rewrite the expression.

$5 y + 1 = {(3 y + 1)}^{1}$

Simplify.

$5 y + 1 = 3 y + 1$

Step 3

Solve for $y$ .

Move all terms containing $y$ to the left side of the equation.

Subtract $3 y$ from both sides of the equation.

$5 y + 1 - 3 y = 1$

Subtract $3 y$ from $5 y$ .

$2 y + 1 = 1$

Move all terms not containing $y$ to the right side of the equation.

Subtract $1$ from both sides of the equation.

$2 y = 1 - 1$

Subtract $1$ from $1$ .

$2 y = 0$

Divide each term in $2 y = 0$ by $2$ and simplify.

Divide each term in $2 y = 0$ by $2$ .

$\frac{2 y}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{0}{2}$

Divide $y$ by $1$ .

$y = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$y = 0$

Solve the Rational Equation for y square root of 5y+1 = square root of 3y+1

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