Solve the equations `3x+5y=9 and 2x+3y=4` by cross multiplication method.

Answer

To solve the equations \(3x + 5y = 9\) and \(2x + 3y = 4\) using the cross multiplication method, follow these steps:

### Step 1: Rewrite the equations in standard form
We can express both equations in the form \(Ax + By + C = 0\).

1. For the first equation:
\[ 3x + 5y - 9 = 0 \]
This gives us \(A_1 = 3\), \(B_1 = 5\), and \(C_1 = -9\).

2. For the second equation:
\[ 2x + 3y - 4 = 0 \]
This gives us \(A_2 = 2\), \(B_2 = 3\), and \(C_2 = -4\).

### Step 2: Set up the cross multiplication
We will set up the cross multiplication using the coefficients from the equations:

\[ \frac{x}{B_1C_2 - B_2C_1} = \frac{y}{A_2C_1 - A_1C_2} = \frac{1}{A_1B_2 - A_2B_1} \]

Substituting the values:
- \(B_1 = 5\), \(B_2 = 3\), \(C_1 = -9\), \(C_2 = -4\)
- \(A_1 = 3\), \(A_2 = 2\)

### Step 3: Calculate the determinants
1. Calculate \(B_1C_2 - B_2C_1\):
\[ 5 \cdot (-4) - 3 \cdot (-9) = -20 + 27 = 7 \]

2. Calculate \(A_2C_1 - A_1C_2\):
\[ 2 \cdot (-9) - 3 \cdot (-4) = -18 + 12 = -6 \]

3. Calculate \(A_1B_2 - A_2B_1\):
\[ 3 \cdot 3 - 2 \cdot 5 = 9 - 10 = -1 \]

### Step 4: Set up the equations
Now we can set up the equations from the determinants:
\[ \frac{x}{7} = \frac{y}{-6} = \frac{1}{-1} \]

### Step 5: Solve for \(x\) and \(y\)
1. From \(\frac{x}{7} = -1\):
\[ x = -1 \cdot 7 = -7 \]

2. From \(\frac{y}{-6} = -1\):
\[ y = -1 \cdot -6 = 6 \]

### Final Solution
The solution to the equations is:
\[ x = -7, \quad y = 6 \]