Solve : `(4-3x)/(5)+(7-x)/(3)+4(1)/(3)=0`.
Hence, find the value of 'p', if `3p-2x+1=0`.

Answer

To solve the equation \(\frac{4-3x}{5} + \frac{7-x}{3} + 4\frac{1}{3} = 0\), we will follow these steps:

### Step 1: Convert Mixed Number to Improper Fraction
First, we convert \(4\frac{1}{3}\) into an improper fraction:
\[ 4\frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3} \]
So, the equation becomes:
\[ \frac{4-3x}{5} + \frac{7-x}{3} + \frac{13}{3} = 0 \]

### Step 2: Combine the Fractions
Next, we need to find a common denominator for the fractions. The least common multiple (LCM) of 5 and 3 is 15. We rewrite each term with a denominator of 15:
\[ \frac{4-3x}{5} = \frac{3(4-3x)}{15} = \frac{12 - 9x}{15} \]
\[ \frac{7-x}{3} = \frac{5(7-x)}{15} = \frac{35 - 5x}{15} \]
\[ \frac{13}{3} = \frac{5 \times 13}{15} = \frac{65}{15} \]

Now, substituting these back into the equation gives:
\[ \frac{12 - 9x + 35 - 5x + 65}{15} = 0 \]

### Step 3: Combine the Numerators
Combine the numerators:
\[ 12 - 9x + 35 - 5x + 65 = 0 \]
Combine like terms:
\[ (12 + 35 + 65) + (-9x - 5x) = 0 \]
\[ 112 - 14x = 0 \]

### Step 4: Solve for \(x\)
Now, we can solve for \(x\):
\[ -14x = -112 \]
\[ x = \frac{-112}{-14} = 8 \]

### Step 5: Substitute \(x\) into the Second Equation
Now that we have \(x = 8\), we substitute this value into the second equation \(3p - 2x + 1 = 0\):
\[ 3p - 2(8) + 1 = 0 \]
\[ 3p - 16 + 1 = 0 \]
\[ 3p - 15 = 0 \]

### Step 6: Solve for \(p\)
Now, solve for \(p\):
\[ 3p = 15 \]
\[ p = \frac{15}{3} = 5 \]

### Final Answer
Thus, the value of \(p\) is \(5\).

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