Question

solve and check the linear equation.
$25 - \frac{x}{3} = \frac{x}{2}$
what is the solution? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. the solution set is \\{\\}.
\\(\bigcirc\\) b. the solution set is \\{x | x is a real number\\}.
\\(\bigcirc\\) c. the solution set is \\(\varnothing\\)

Explanation:

Step1: Add $\frac{x}{3}$ to both sides

To isolate the terms with \( x \), we add \(\frac{x}{3}\) to both sides of the equation \( 25 - \frac{x}{3} = \frac{x}{2} \). This gives us \( 25 = \frac{x}{2} + \frac{x}{3} \).

Step2: Find a common denominator for the right side

The common denominator of 2 and 3 is 6. So we rewrite \(\frac{x}{2}\) as \(\frac{3x}{6}\) and \(\frac{x}{3}\) as \(\frac{2x}{6}\). Then the equation becomes \( 25 = \frac{3x}{6} + \frac{2x}{6} \).

Step3: Combine like terms on the right side

Adding \(\frac{3x}{6}\) and \(\frac{2x}{6}\) gives \(\frac{5x}{6}\). So now our equation is \( 25 = \frac{5x}{6} \).

Step4: Solve for \( x \)

Multiply both sides of the equation by \(\frac{6}{5}\) to isolate \( x \). So \( x = 25\times\frac{6}{5} \). Calculating that, \( 25\div5 = 5 \), and \( 5\times6 = 30 \). So \( x = 30 \).

Step5: Check the solution

Substitute \( x = 30 \) back into the original equation. Left side: \( 25 - \frac{30}{3} = 25 - 10 = 15 \). Right side: \( \frac{30}{2} = 15 \). Both sides are equal, so \( x = 30 \) is a valid solution.

Answer:

A. The solution set is \(\{30\}\)