Let's solve the equation step by step:

The equation is:

\[
\frac{1}{3}x + \frac{5}{3} = \frac{5}{16}x - 2
\]

### Step 1: Eliminate the fractions
To eliminate the fractions, multiply every term in the equation by the least common multiple (LCM) of the denominators. The denominators are 3 and 16, and the LCM of 3 and 16 is 48.

Multiply the entire equation by 48:

\[
48\left(\frac{1}{3}x\right) + 48\left(\frac{5}{3}\right) = 48\left(\frac{5}{16}x\right) - 48(2)
\]

Simplify:

\[
16x + 80 = 15x - 96
\]

### Step 2: Move all \(x\) terms to one side
To move the \(x\) terms to one side, subtract \(15x\) from both sides:

\[
16x - 15x + 80 = -96
\]

Simplify:

\[
x + 80 = -96
\]

### Step 3: Isolate \(x\)
Subtract 80 from both sides:

\[
x = -96 - 80
\]

Simplify:

\[
x = -176
\]

### Solution:
The value of \(x\) is \(-176\).

Do you want more details on any step, or do you have any questions?

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Here are some related questions you might find helpful:

1. How do you find the least common multiple of two numbers?
2. Why do we multiply by the LCM to eliminate fractions in an equation?
3. How can you check your solution to an equation?
4. What if both sides of the equation have different coefficients for \(x\)?
5. What are linear equations, and how are they different from quadratic equations?
6. How does changing the sign of a number affect an equation?
7. What is the significance of isolating the variable in an equation?
8. How do you graph the solution to a linear equation in one variable?

**Tip:** When dealing with equations that involve fractions, clearing the fractions early by multiplying by the LCM can simplify the problem and reduce errors.

Learn how to solve the linear equation 1/3x + 5/3 = 5/16x - 2 step-by-step. This problem involves solving equations with fractions, suitable for students in Grades 7-9.

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