Solve following pairs of linear equations using cross-multiplication method :
`{:(5x-3y=2),(4x+7y=-3):}`

Answer

To solve the given pair of linear equations using the cross-multiplication method, we follow these steps:

### Step 1: Write the equations in standard form
We start with the given equations:
1. \( 5x - 3y = 2 \)
2. \( 4x + 7y = -3 \)

We need to rearrange them into the standard form \( Ax + By + C = 0 \).

For the first equation:
\[ 5x - 3y - 2 = 0 \]
Let's name this as Equation (1).

For the second equation:
\[ 4x + 7y + 3 = 0 \]
Let's name this as Equation (2).

### Step 2: Identify coefficients
From the equations, we can identify the coefficients:
- From Equation (1): \( A_1 = 5, B_1 = -3, C_1 = -2 \)
- From Equation (2): \( A_2 = 4, B_2 = 7, C_2 = 3 \)

### Step 3: Set up the cross-multiplication
Using the cross-multiplication method, we set up the following relationships:
\[ \frac{x}{B_1 C_2 - B_2 C_1} = \frac{y}{C_1 A_2 - A_1 C_2} = \frac{1}{A_1 B_2 - A_2 B_1} \]

### Step 4: Calculate the determinants
Now we calculate each determinant:

1. For \( B_1 C_2 - B_2 C_1 \):
\[ B_1 C_2 - B_2 C_1 = (-3)(3) - (7)(-2) = -9 + 14 = 5 \]

2. For \( C_1 A_2 - A_1 C_2 \):
\[ C_1 A_2 - A_1 C_2 = (-2)(4) - (5)(3) = -8 - 15 = -23 \]

3. For \( A_1 B_2 - A_2 B_1 \):
\[ A_1 B_2 - A_2 B_1 = (5)(7) - (4)(-3) = 35 + 12 = 47 \]

### Step 5: Set up the equations
Now we can write the equations based on the determinants:
\[ \frac{x}{5} = \frac{y}{-23} = \frac{1}{47} \]

### Step 6: Solve for \( x \) and \( y \)
From the first part:
\[ \frac{x}{5} = \frac{1}{47} \implies x = \frac{5}{47} \]

From the second part:
\[ \frac{y}{-23} = \frac{1}{47} \implies y = -\frac{23}{47} \]

### Final Solution
Thus, the solution to the equations is:
\[ x = \frac{5}{47}, \quad y = -\frac{23}{47} \]

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