Solve:`(5x-4)/8-(x-3)/5=(x+6)/4`

Options

A. `x=7`
B. `x=8`
C. `x=6`
D. `x=9`

Answer

To solve the equation \(\frac{5x - 4}{8} - \frac{x - 3}{5} = \frac{x + 6}{4}\), we will follow these steps:

### Step 1: Eliminate the fractions
To eliminate the fractions, we can find the least common multiple (LCM) of the denominators (8, 5, and 4). The LCM of 8, 5, and 4 is 40. We will multiply every term in the equation by 40.

\[ 40 \left(\frac{5x - 4}{8}\right) - 40 \left(\frac{x - 3}{5}\right) = 40 \left(\frac{x + 6}{4}\right) \]

### Step 2: Simplify each term
Now, we simplify each term:

1. \(40 \cdot \frac{5x - 4}{8} = 5 \cdot (5x - 4) = 25x - 20\)
2. \(40 \cdot \frac{x - 3}{5} = 8 \cdot (x - 3) = 8x - 24\)
3. \(40 \cdot \frac{x + 6}{4} = 10 \cdot (x + 6) = 10x + 60\)

So, the equation becomes:

\[ 25x - 20 - (8x - 24) = 10x + 60 \]

### Step 3: Distribute and combine like terms
Distributing the negative sign in the left-hand side:

\[ 25x - 20 - 8x + 24 = 10x + 60 \]

Now combine like terms on the left side:

\[ (25x - 8x) + (-20 + 24) = 10x + 60 \]

This simplifies to:

\[ 17x + 4 = 10x + 60 \]

### Step 4: Isolate the variable
Next, we want to isolate \(x\). We can do this by moving \(10x\) to the left side and \(4\) to the right side:

\[ 17x - 10x = 60 - 4 \]

This simplifies to:

\[ 7x = 56 \]

### Step 5: Solve for \(x\)
Now, divide both sides by 7:

\[ x = \frac{56}{7} = 8 \]

### Final Answer
Thus, the solution to the equation is:

\[ \boxed{8} \]

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