The value of x and y from the equation `x-y=3 and (x)/(3)+(y)/(2)=6` are

Options

A. `x=6, y=9`
B. `x=9,y=6`
C. `x=8,y=5`
D. None of these

Answer

To solve the equations \( x - y = 3 \) and \( \frac{x}{3} + \frac{y}{2} = 6 \), we will follow these steps:

### Step 1: Write down the equations
We have the following two equations:
1. \( x - y = 3 \) (Equation 1)
2. \( \frac{x}{3} + \frac{y}{2} = 6 \) (Equation 2)

### Step 2: Eliminate fractions in Equation 2
To eliminate the fractions in Equation 2, we can find the least common multiple (LCM) of the denominators (3 and 2), which is 6. We will multiply the entire equation by 6:

\[ 6 \left( \frac{x}{3} + \frac{y}{2} \right) = 6 \cdot 6 \]

This simplifies to:

\[ 2x + 3y = 36 \quad \text{(Equation 3)} \]

### Step 3: Solve Equation 1 for one variable
From Equation 1, we can express \( x \) in terms of \( y \):

\[ x = y + 3 \]

### Step 4: Substitute \( x \) in Equation 3
Now, we will substitute \( x \) from Equation 1 into Equation 3:

\[ 2(y + 3) + 3y = 36 \]

### Step 5: Simplify and solve for \( y \)
Expanding the equation gives:

\[ 2y + 6 + 3y = 36 \]

Combining like terms:

\[ 5y + 6 = 36 \]

Now, subtract 6 from both sides:

\[ 5y = 30 \]

Dividing by 5:

\[ y = 6 \]

### Step 6: Substitute \( y \) back to find \( x \)
Now that we have \( y \), we can substitute it back into the equation we derived for \( x \):

\[ x = y + 3 = 6 + 3 = 9 \]

### Final Solution
Thus, the values of \( x \) and \( y \) are:

\[ x = 9, \quad y = 6 \]

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