Solve: `2/3x+3/2x-3/4x=17/4`

Answer

To solve the equation \( \frac{2}{3}x + \frac{3}{2}x - \frac{3}{4}x = \frac{17}{4} \), we will follow these steps:

### Step 1: Find a common denominator for the fractions on the left side.
The denominators are 3, 2, and 4. The least common multiple (LCM) of these numbers is 12.

### Step 2: Rewrite each term with the common denominator.
We will convert each fraction to have a denominator of 12:
- \( \frac{2}{3}x = \frac{2 \times 4}{3 \times 4}x = \frac{8}{12}x \)
- \( \frac{3}{2}x = \frac{3 \times 6}{2 \times 6}x = \frac{18}{12}x \)
- \( \frac{3}{4}x = \frac{3 \times 3}{4 \times 3}x = \frac{9}{12}x \)

Now, we can rewrite the equation:
\[ \frac{8}{12}x + \frac{18}{12}x - \frac{9}{12}x = \frac{17}{4} \]

### Step 3: Combine the fractions on the left side.
Combine the fractions:
\[ \frac{8 + 18 - 9}{12}x = \frac{17}{4} \]
Calculating the numerator:
\[ 8 + 18 - 9 = 17 \]
So, we have:
\[ \frac{17}{12}x = \frac{17}{4} \]

### Step 4: Eliminate the fraction by multiplying both sides by the denominator.
To eliminate the fraction, we can multiply both sides by 12:
\[ 17x = 12 \times \frac{17}{4} \]

### Step 5: Simplify the right side.
Calculating the right side:
\[ 12 \times \frac{17}{4} = \frac{12 \times 17}{4} = 3 \times 17 = 51 \]
So, we have:
\[ 17x = 51 \]

### Step 6: Solve for \( x \).
Now, divide both sides by 17:
\[ x = \frac{51}{17} = 3 \]

### Final Answer:
Thus, the solution to the equation is:
\[ x = 3 \]

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