Sure, let's solve the equation step-by-step:
We are given the equation:
[tex]\[
\frac{64x}{7} - \frac{7}{x} = 0
\][/tex]
Step 1: Eliminate the fractions by multiplying both sides of the equation by [tex]\(7x\)[/tex]:
[tex]\[
7x \left(\frac{64x}{7} - \frac{7}{x}\right) = 7x \cdot 0
\][/tex]
Step 2: Simplify the resulting equation:
[tex]\[
64x^2 - 7 \cdot 7 = 0
\][/tex]
[tex]\[
64x^2 - 49 = 0
\][/tex]
Step 3: Move the constant term to the other side of the equation:
[tex]\[
64x^2 = 49
\][/tex]
Step 4: Divide both sides by 64 to isolate [tex]\(x^2\)[/tex]:
[tex]\[
x^2 = \frac{49}{64}
\][/tex]
Step 5: Take the square root of both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \pm \sqrt{\frac{49}{64}}
\][/tex]
Step 6: Simplify the square root:
[tex]\[
x = \pm \frac{7}{8}
\][/tex]
Thus, the solutions to the equation [tex]\(\frac{64x}{7} - \frac{7}{x} = 0\)[/tex] are:
[tex]\[
x = -\frac{7}{8} \quad \text{and} \quad x = \frac{7}{8}
\][/tex]
