Solve the equation $$ \frac{10}{x}=\frac{-4}{3 x}+6 $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\frac{10}{x} = \frac{-4}{3x} + 6
$$

follow these steps:

### Step 1: Eliminate the Fractions

First, identify the least common denominator (LCD) to eliminate the fractions. The denominators here are $$ x $$ and $$ 3x $$. The LCD is $$ 3x $$.

Multiply **both sides** of the equation by $$ 3x $$:

$$
3x \cdot \frac{10}{x} = 3x \cdot \left( \frac{-4}{3x} \right) + 3x \cdot 6
$$

### Step 2: Simplify Each Term

Let's simplify each term after multiplication:

1. **First Term:**
   $$
   3x \cdot \frac{10}{x} = 30
   $$
   (The $$ x $$ cancels out.)

2. **Second Term:**
   $$
   3x \cdot \frac{-4}{3x} = -4
   $$
   (The $$ 3x $$ cancels out.)

3. **Third Term:**
   $$
   3x \cdot 6 = 18x
   $$

After simplification, the equation becomes:

$$
30 = -4 + 18x
$$

### Step 3: Solve for $$ x $$

Now, isolate $$ x $$:

1. **Add 4** to both sides to get rid of the constant term on the right:

$$
   30 + 4 = 18x
   $$
   
   $$
   34 = 18x
   $$

2. **Divide both sides** by 18 to solve for $$ x $$:

$$
   x = \frac{34}{18}
   $$

3. **Simplify the fraction** by dividing numerator and denominator by 2:

$$
   x = \frac{17}{9}
   $$

### Final Answer

$$
x = \frac{17}{9}
$$

Alternatively, as a decimal:

$$
x \approx 1.89
$$
$$ x = \frac{17}{9} $$

Solve the equation \( \frac{10}{x}=\frac{-4}{3 x}+6 \)