Let's break down and solve this equation step by step:

### The given equation is:
\[
5(2) + 2(3)x + \frac{1}{6}x = 2x - 2
\]

#### Step 1: Simplify both sides

- First, simplify \(5(2)\):
\[
5(2) = 10
\]
So, the equation becomes:
\[
10 + 6x + \frac{1}{6}x = 2x - 2
\]

#### Step 2: Combine like terms

- Add \(6x\) and \(\frac{1}{6}x\) on the left-hand side:
\[
6x + \frac{1}{6}x = \frac{36}{6}x + \frac{1}{6}x = \frac{37}{6}x
\]
Now the equation looks like this:
\[
10 + \frac{37}{6}x = 2x - 2
\]

#### Step 3: Eliminate the fraction

To eliminate the fraction, multiply the entire equation by 6:
\[
6 \times \left(10 + \frac{37}{6}x\right) = 6 \times (2x - 2)
\]
This gives:
\[
60 + 37x = 12x - 12
\]

#### Step 4: Move the variables to one side

Subtract \(12x\) from both sides:
\[
60 + 37x - 12x = -12
\]
This simplifies to:
\[
60 + 25x = -12
\]

#### Step 5: Solve for \(x\)

Subtract 60 from both sides:
\[
25x = -12 - 60
\]
\[
25x = -72
\]
Now, divide by 25:
\[
x = \frac{-72}{25}
\]

Thus, the solution is:
\[
x = \frac{-72}{25}
\]

### Final Answer:
\[
x = \frac{-72}{25}
\]

Would you like further details or clarifications?

Here are 5 questions that expand on this solution:
1. How do you handle equations with multiple fractions?
2. What steps are crucial in combining terms with different coefficients?
3. How does multiplying by a common denominator simplify solving equations?
4. Why is it important to move variables to one side of the equation?
5. How do you interpret negative solutions in real-world problems?

**Tip**: When working with fractions in equations, multiplying through by the least common denominator (LCD) can simplify your work!

Learn how to solve the linear equation 5(2) + 2(3)x + 1/6x = 2x – 2. This problem involves combining like terms, solving equations with fractions, and simplifying expressions step by step. Suitable for students in Grades 8-10.

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