Question

solve the equation, and check the solution.
\\(\frac{3}{5}t - \frac{1}{10}t = t - \frac{7}{2}\\)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. there is exactly one solution. the solution set is \\(\\{\square\\}\\).
(simplify your answer.)
\\(\bigcirc\\) b. the solution set is \\{all real numbers\\}.
\\(\bigcirc\\) c. the solution set is \\(\varnothing\\).

Explanation:

Step1: Simplify left - hand side

First, we simplify the left - hand side of the equation \(\frac{3}{5}t-\frac{1}{10}t\). We find a common denominator for the two fractions. The common denominator of 5 and 10 is 10. So we rewrite \(\frac{3}{5}t\) as \(\frac{6}{10}t\). Then \(\frac{6}{10}t-\frac{1}{10}t=\frac{6 - 1}{10}t=\frac{5}{10}t=\frac{1}{2}t\). The equation becomes \(\frac{1}{2}t=t-\frac{7}{2}\).

Step2: Isolate the variable \(t\)

Subtract \(\frac{1}{2}t\) from both sides of the equation: \(\frac{1}{2}t-\frac{1}{2}t=t-\frac{1}{2}t-\frac{7}{2}\). The left - hand side is 0, and the right - hand side simplifies to \(\frac{2t - t}{2}-\frac{7}{2}=\frac{t}{2}-\frac{7}{2}\). So we have \(0=\frac{t}{2}-\frac{7}{2}\). Then add \(\frac{7}{2}\) to both sides: \(\frac{7}{2}=\frac{t}{2}\).

Step3: Solve for \(t\)

Multiply both sides of the equation \(\frac{7}{2}=\frac{t}{2}\) by 2 to get \(t = 7\).

Step4: Check the solution

Substitute \(t = 7\) into the original equation.
Left - hand side: \(\frac{3}{5}\times7-\frac{1}{10}\times7=\frac{21}{5}-\frac{7}{10}=\frac{42 - 7}{10}=\frac{35}{10}=\frac{7}{2}\)
Right - hand side: \(7-\frac{7}{2}=\frac{14 - 7}{2}=\frac{7}{2}\)
Since the left - hand side equals the right - hand side when \(t = 7\), the solution is valid.

Answer:

A. There is exactly one solution. The solution set is \(\{7\}\)