Solve the Rational Equation for x -5 = square root of 3x-7

$- 5 = \sqrt{3 x - 7}$

Step 1

Rewrite the equation as $\sqrt{3 x - 7} = - 5$ .

$\sqrt{3 x - 7} = - 5$

Step 2

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

${\sqrt{3 x - 7}}^{2} = {(- 5)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x - 7}$ as ${(3 x - 7)}^{\frac{1}{2}}$ .

${({(3 x - 7)}^{\frac{1}{2}})}^{2} = {(- 5)}^{2}$

Simplify the left side.

Simplify ${({(3 x - 7)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 x - 7)}^{\frac{1}{2}})}^{2}$ .

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

${(3 x - 7)}^{\frac{1}{2} \cdot 2} = {(- 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 x - 7)}^{\frac{1}{2} \cdot 2} = {(- 5)}^{2}$

Rewrite the expression.

${(3 x - 7)}^{1} = {(- 5)}^{2}$

Simplify.

$3 x - 7 = {(- 5)}^{2}$

Simplify the right side.

Raise $- 5$ to the power of $2$ .

$3 x - 7 = 25$

Step 4

Solve for $x$ .

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

Add $7$ to both sides of the equation.

$3 x = 25 + 7$

Add $25$ and $7$ .

$3 x = 32$

Divide each term in $3 x = 32$ by $3$ and simplify.

Divide each term in $3 x = 32$ by $3$ .

$\frac{3 x}{3} = \frac{32}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{32}{3}$

Divide $x$ by $1$ .

$x = \frac{32}{3}$

Step 5

Exclude the solutions that do not make $- 5 = \sqrt{3 x - 7}$ true.

$No solution$