If `7x + 6 y = 5xy and 10 x - 4y = 4xy,` then value of x and y is

Options

A. `3,2`
B. `2,3`
C. `4,2`
D. `5,6`

Answer

To solve the equations \(7x + 6y = 5xy\) and \(10x - 4y = 4xy\), we will follow these steps:

### Step 1: Rewrite the equations
The given equations are:
1. \(7x + 6y = 5xy\)
2. \(10x - 4y = 4xy\)

### Step 2: Divide both sides of the first equation by \(xy\)
From the first equation:
\[ \frac{7x}{xy} + \frac{6y}{xy} = \frac{5xy}{xy} \]
This simplifies to:
\[ \frac{7}{y} + \frac{6}{x} = 5 \]
Let's call this Equation (1).

### Step 3: Divide both sides of the second equation by \(xy\)
From the second equation:
\[ \frac{10x}{xy} - \frac{4y}{xy} = \frac{4xy}{xy} \]
This simplifies to:
\[ \frac{10}{y} - \frac{4}{x} = 4 \]
Let's call this Equation (2).

### Step 4: Multiply Equation (1) by 10
Multiplying Equation (1) by 10 gives:
\[ 10 \left(\frac{7}{y} + \frac{6}{x}\right) = 10 \cdot 5 \]
This simplifies to:
\[ \frac{70}{y} + \frac{60}{x} = 50 \]
Let's call this Equation (3).

### Step 5: Multiply Equation (2) by 7
Multiplying Equation (2) by 7 gives:
\[ 7 \left(\frac{10}{y} - \frac{4}{x}\right) = 7 \cdot 4 \]
This simplifies to:
\[ \frac{70}{y} - \frac{28}{x} = 28 \]
Let's call this Equation (4).

### Step 6: Subtract Equation (4) from Equation (3)
Now we subtract Equation (4) from Equation (3):
\[ \left(\frac{70}{y} + \frac{60}{x}\right) - \left(\frac{70}{y} - \frac{28}{x}\right) = 50 - 28 \]
This simplifies to:
\[ \frac{60}{x} + \frac{28}{x} = 22 \]
Combining the terms gives:
\[ \frac{88}{x} = 22 \]

### Step 7: Solve for \(x\)
Now, we can solve for \(x\):
\[ x = \frac{88}{22} = 4 \]

### Step 8: Substitute \(x\) back into Equation (1) to find \(y\)
Now substitute \(x = 4\) back into Equation (1):
\[ \frac{7}{y} + \frac{6}{4} = 5 \]
This simplifies to:
\[ \frac{7}{y} + \frac{3}{2} = 5 \]
Subtract \(\frac{3}{2}\) from both sides:
\[ \frac{7}{y} = 5 - \frac{3}{2} = \frac{10}{2} - \frac{3}{2} = \frac{7}{2} \]
Now, cross-multiply to solve for \(y\):
\[ 7 \cdot 2 = 7y \implies 14 = 7y \implies y = 2 \]

### Final Answer
Thus, the values of \(x\) and \(y\) are:
\[ x = 4, \quad y = 2 \]