Let's solve the given linear equation step-by-step to find the value of [tex]\(x\)[/tex]:
[tex]\[
\frac{2}{3} x - \frac{1}{2} = \frac{1}{3} + \frac{5}{6} x
\][/tex]
### Step 1: Eliminate the fractions
To make the equation easier to work with, we can start by eliminating the fractions. The common denominator for the fractions [tex]\(\frac{2}{3}\)[/tex], [tex]\(\frac{1}{2}\)[/tex], [tex]\(\frac{1}{3}\)[/tex], and [tex]\(\frac{5}{6}\)[/tex] is 6. We can multiply every term in the equation by 6 to clear the fractions:
[tex]\[
6 \left(\frac{2}{3} x\right) - 6 \left(\frac{1}{2}\right) = 6 \left(\frac{1}{3}\right) + 6 \left(\frac{5}{6} x\right)
\][/tex]
This simplifies to:
[tex]\[
4x - 3 = 2 + 5x
\][/tex]
### Step 2: Rearrange the equation
Next, we want to get all the [tex]\(x\)[/tex] terms on one side of the equation and the constant terms on the other side. We can start by subtracting [tex]\(4x\)[/tex] from both sides:
[tex]\[
4x - 3 - 4x = 2 + 5x - 4x
\][/tex]
This simplifies to:
[tex]\[
-3 = 2 + x
\][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
Now, we isolate [tex]\(x\)[/tex] by subtracting 2 from both sides of the equation:
[tex]\[
-3 - 2 = x
\][/tex]
This simplifies to:
[tex]\[
x = -5
\][/tex]
### Step 4: Verify the solution
To ensure our solution is correct, we can substitute [tex]\(x = -5\)[/tex] back into the original equation:
[tex]\[
\frac{2}{3}(-5) - \frac{1}{2} = \frac{1}{3} + \frac{5}{6}(-5)
\][/tex]
Simplifying each term:
[tex]\[
-\frac{10}{3} - \frac{1}{2} = \frac{1}{3} - \frac{25}{6}
\][/tex]
Convert [tex]\(-\frac{10}{3}\)[/tex], [tex]\(\frac{1}{2}\)[/tex], and [tex]\(\frac{1}{3}\)[/tex], [tex]\(\frac{25}{6}\)[/tex] to a common denominator (which is 6 in this case):
[tex]\[
- \frac{20}{6} - \frac{3}{6} = \frac{2}{6} - \frac{25}{6}
\][/tex]
Combine the fractions:
[tex]\[
- \frac{23}{6} = - \frac{23}{6}
\][/tex]
Both sides are equal, confirming our solution is correct. Thus, the solution to the linear equation is:
[tex]\[
\boxed{-5}
\][/tex]
