To solve the equation [tex]\(\sqrt{9x - 28} = \sqrt{5x}\)[/tex], follow these steps:
1. Square Both Sides:
First, eliminate the square roots by squaring both sides of the equation.
[tex]\[
(\sqrt{9x - 28})^2 = (\sqrt{5x})^2
\][/tex]
This simplifies to:
[tex]\[
9x - 28 = 5x
\][/tex]
2. Rearrange the Equation:
Move all terms involving [tex]\(x\)[/tex] to one side of the equation and the constant terms to the other side.
[tex]\[
9x - 5x = 28
\][/tex]
Simplify the left-hand side:
[tex]\[
4x = 28
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Divide both sides by 4 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{28}{4}
\][/tex]
Simplify the division:
[tex]\[
x = 7
\][/tex]
4. Check for Validity:
Substitute [tex]\(x = 7\)[/tex] back into the original equation to ensure it is not an extraneous solution.
The left side of the equation:
[tex]\[
\sqrt{9(7) - 28} = \sqrt{63 - 28} = \sqrt{35}
\][/tex]
The right side of the equation:
[tex]\[
\sqrt{5(7)} = \sqrt{35}
\][/tex]
Since both sides are equal, the solution [tex]\(x = 7\)[/tex] is valid.
Therefore, the solution to the equation [tex]\(\sqrt{9x - 28} = \sqrt{5x}\)[/tex] is:
[tex]\[
x = 7
\][/tex]
