Question

In the following exercises, solve:

$\sqrt{3u-2}=\sqrt{5u+1}$.

Short answer

The solution is,$u=-\frac{3}{2}$.

Step-by-step solution

Step 1. Given Information.

The expression is,

$\sqrt{3u-2}=\sqrt{5u+1}$.

Step 2. Finding the solution.

$\sqrt{3u-2}=\sqrt{5u+1}$

Squaring both sides,

role="math" localid="1647890790168" $$\begin{gathered} \Rightarrow {\left(\sqrt{3u-2}\right)}^2={\left(\sqrt{5u+1}\right)}^2 \\ \Rightarrow 3u-2=5u+1 \end{gathered}$$

Adding $2$on both sides,

$$\begin{gathered} \Rightarrow 3u=5u+1+2 \\ \Rightarrow 3u=5u+3 \end{gathered}$$

Subtracting $5u$on both sides,

$$\begin{gathered} \Rightarrow 3u-5u=3 \\ \Rightarrow -2u=3 \end{gathered}$$

Dividing both sides by $(-2)$,

$$\begin{gathered} \Rightarrow \frac{-2u}{-2}=\frac{3}{-2} \\ \Rightarrow u=-\frac{3}{2} \end{gathered}$$

Step 3. Checking the solution.

$\sqrt{3u-2}$

Inputting $u=-\frac{3}{2}$,

$$\begin{gathered} =\sqrt{3\times (-\frac{3}{2})-2} \\ =\sqrt{-\frac{9}{2}-2} \\ =\sqrt{\frac{-9-4}{2}} \\ =\sqrt{-\frac{13}{2}} \end{gathered}$$

$\sqrt{5u+1}$

Inputting $u=-\frac{3}{2}$,

$$\begin{gathered} =\sqrt{5\times (-\frac{3}{2})+1} \\ =\sqrt{-\frac{15}{2}+1} \\ =\sqrt{\frac{-15+2}{2}} \\ =\sqrt{-\frac{13}{2}} \end{gathered}$$

The solution is true.