Solve : `(5x -2)/(3) + (4x+3)/(2) = (3x+19)/(2)`

Answer

To solve the equation \((5x - 2)/3 + (4x + 3)/2 = (3x + 19)/2\), we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
The denominators in the equation are 3 and 2. The LCM of 3 and 2 is 6.

### Step 2: Rewrite Each Fraction with the LCM
We will rewrite each term in the equation with the LCM of 6 as the denominator:

\[ \frac{5x - 2}{3} = \frac{2(5x - 2)}{6} \]
\[ \frac{4x + 3}{2} = \frac{3(4x + 3)}{6} \]
\[ \frac{3x + 19}{2} = \frac{3(3x + 19)}{6} \]

### Step 3: Substitute Back into the Equation
Now, substituting these back into the equation gives us:

\[ \frac{2(5x - 2) + 3(4x + 3)}{6} = \frac{3(3x + 19)}{6} \]

### Step 4: Eliminate the Denominator
Since both sides of the equation have the same denominator (6), we can multiply through by 6 to eliminate it:

\[ 2(5x - 2) + 3(4x + 3) = 3(3x + 19) \]

### Step 5: Distribute
Now we will distribute on both sides:

Left side:
\[ 2(5x) - 2(2) + 3(4x) + 3(3) = 10x - 4 + 12x + 9 \]
Combining like terms gives us:
\[ (10x + 12x) + (-4 + 9) = 22x + 5 \]

Right side:
\[ 3(3x) + 3(19) = 9x + 57 \]

So now we have:
\[ 22x + 5 = 9x + 57 \]

### Step 6: Move All Terms Involving \(x\) to One Side
Subtract \(9x\) from both sides:
\[ 22x - 9x + 5 = 57 \]
This simplifies to:
\[ 13x + 5 = 57 \]

### Step 7: Move Constant to the Other Side
Subtract 5 from both sides:
\[ 13x = 57 - 5 \]
This simplifies to:
\[ 13x = 52 \]

### Step 8: Solve for \(x\)
Now, divide both sides by 13:
\[ x = \frac{52}{13} \]
This simplifies to:
\[ x = 4 \]

### Final Answer
The value of \(x\) is \(4\).

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