Solve , Uing cross multiplication method
`{:(4x-y=5),(5y-4x=7):}`

Answer

To solve the system of equations using the cross multiplication method, we have the following equations:

1. \( 4x - y = 5 \)
2. \( 5y - 4x = 7 \)

### Step 1: Rearranging the Equations
First, we need to rearrange both equations into the standard form \( Ax + By + C = 0 \).

For the first equation:
\[ 4x - y - 5 = 0 \quad \text{(Equation 1)} \]

For the second equation:
\[ -4x + 5y - 7 = 0 \quad \text{(Equation 2)} \]

### Step 2: Identifying Coefficients
Now, we identify the coefficients from the equations:

From Equation 1:
- \( a_1 = 4 \)
- \( b_1 = -1 \)
- \( c_1 = -5 \)

From Equation 2:
- \( a_2 = -4 \)
- \( b_2 = 5 \)
- \( c_2 = -7 \)

### Step 3: Applying the Cross Multiplication Formula
The cross multiplication method gives us the following relationships:
\[ \frac{x}{b_1 c_2 - b_2 c_1} = \frac{y}{c_1 a_2 - c_2 a_1} = \frac{1}{a_1 b_2 - a_2 b_1} \]

#### Calculating \( b_1 c_2 - b_2 c_1 \):
\[ b_1 c_2 - b_2 c_1 = (-1)(-7) - (5)(-5) = 7 + 25 = 32 \]

#### Calculating \( c_1 a_2 - c_2 a_1 \):
\[ c_1 a_2 - c_2 a_1 = (-5)(-4) - (-7)(4) = 20 + 28 = 48 \]

#### Calculating \( a_1 b_2 - a_2 b_1 \):
\[ a_1 b_2 - a_2 b_1 = (4)(5) - (-4)(-1) = 20 - 4 = 16 \]

### Step 4: Setting Up the Equations
Now we can set up the equations based on the cross multiplication results:
\[ \frac{x}{32} = \frac{y}{48} = \frac{1}{16} \]

### Step 5: Solving for \( x \) and \( y \)

#### Solving for \( x \):
\[ \frac{x}{32} = \frac{1}{16} \implies x = 32 \cdot \frac{1}{16} = 2 \]

#### Solving for \( y \):
\[ \frac{y}{48} = \frac{1}{16} \implies y = 48 \cdot \frac{1}{16} = 3 \]

### Final Answer
The solution to the system of equations is:
\[ x = 2, \quad y = 3 \]