https://file.aiquickdraw.com/mathbot/file/17284339676238d1wo5cj.jpg

The equation provided in the image is:

\[
\frac{1}{3}x + \frac{1}{5} = -2\left(\frac{5}{6}x + 5\right)
\]

Let's go through the steps to solve it:

### Step 1: Distribute the -2 on the right-hand side
\[
-2\left(\frac{5}{6}x + 5\right) = -2 \cdot \frac{5}{6}x - 2 \cdot 5 = -\frac{10}{6}x - 10 = -\frac{5}{3}x - 10
\]

So, the equation becomes:
\[
\frac{1}{3}x + \frac{1}{5} = -\frac{5}{3}x - 10
\]

### Step 2: Eliminate fractions by multiplying the entire equation by the least common denominator (LCD) of 3 and 5, which is 15.

Multiply every term by 15:
\[
15\left(\frac{1}{3}x\right) + 15\left(\frac{1}{5}\right) = 15\left(-\frac{5}{3}x - 10\right)
\]
This simplifies to:
\[
5x + 3 = -25x - 150
\]

### Step 3: Move all terms involving \(x\) to one side and constant terms to the other side.
\[
5x + 25x = -150 - 3
\]
\[
30x = -153
\]

### Step 4: Solve for \(x\)
\[
x = \frac{-153}{30}
\]

Simplify the fraction:
\[
x = \frac{-51}{10}
\]

Thus, the solution is:
\[
x = \frac{-51}{10}
\]

This matches the solution given in the image.

Would you like further details or have any questions?

Here are five related questions:
1. How do you simplify a fraction like \(\frac{-153}{30}\)?
2. What are the steps to clear fractions in an equation?
3. How do you solve equations involving the distributive property?
4. What is the significance of the least common denominator in equations with fractions?
5. How can you check if your solution for \(x\) is correct?

**Tip**: Always try to clear fractions early in solving equations for easier simplification!

Solve the equation: (1/3)x + 1/5 = -2((5/6)x + 5).

Learn how to solve the linear equation (1/3)x + 1/5 = -2((5/6)x + 5) step-by-step. This problem demonstrates key algebraic concepts, including using the distributive property and eliminating fractions by finding the least common denominator. Suitable for students in Grades 8-10.

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