Let's solve the given equation step-by-step:
The equation is:
[tex]\[ \sqrt{\frac{x}{10}} = \sqrt{3x - 58} \][/tex]
1. Square both sides to get rid of the square roots:
[tex]\[ \left(\sqrt{\frac{x}{10}}\right)^2 = \left(\sqrt{3x - 58}\right)^2 \][/tex]
This simplifies to:
[tex]\[ \frac{x}{10} = 3x - 58 \][/tex]
2. Clear the fraction: Multiply both sides by 10 to eliminate the denominator:
[tex]\[ x = 10(3x - 58) \][/tex]
3. Expand and simplify the right-hand side:
[tex]\[ x = 30x - 580 \][/tex]
4. Rearrange the equation to gather all terms involving [tex]\( x \)[/tex] on one side:
[tex]\[ x - 30x = -580 \][/tex]
This simplifies to:
[tex]\[ -29x = -580 \][/tex]
5. Solve for [tex]\( x \)[/tex] by dividing both sides by -29:
[tex]\[ x = \frac{-580}{-29} \][/tex]
[tex]\[ x = 20 \][/tex]
So, the solution to the equation
[tex]\[ \sqrt{\frac{x}{10}} = \sqrt{3x - 58} \][/tex]
is:
[tex]\[ x = 20 \][/tex]
