Question

solve the equation.
2(3t - 2) + t = 12 - t
...
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is {}. (simplify your answer.)
b. the solution set is {all real numbers}
c. the solution set is ∅.

Explanation:

Step1: Expand the left side

First, we expand \(2(3t - 2)\) using the distributive property \(a(b - c)=ab - ac\). So, \(2(3t - 2)=2\times3t - 2\times2 = 6t - 4\). The original equation \(2(3t - 2)+t = 12 - t\) becomes \(6t - 4 + t = 12 - t\).

Step2: Combine like terms on the left side

Combine the \(t\) terms on the left: \(6t + t = 7t\). So the equation is now \(7t - 4 = 12 - t\).

Step3: Add \(t\) to both sides

To get all the \(t\) terms on one side, we add \(t\) to both sides. \(7t - 4 + t = 12 - t + t\), which simplifies to \(8t - 4 = 12\).

Step4: Add 4 to both sides

Now, add 4 to both sides to isolate the term with \(t\). \(8t - 4 + 4 = 12 + 4\), so \(8t = 16\).

Step5: Divide both sides by 8

Divide both sides by 8 to solve for \(t\). \(\frac{8t}{8}=\frac{16}{8}\), which gives \(t = 2\).

Answer:

A. The solution set is \(\{2\}\)