Solve, using cross - multiplication :
8x + 5y = 9
3x + 2y = 4

Answer

To solve the equations using cross-multiplication, we start with the given equations:

1. \( 8x + 5y = 9 \)
2. \( 3x + 2y = 4 \)

### Step 1: Rewrite the equations in standard form
We can rewrite the equations in the form \( ax + by + c = 0 \):

1. \( 8x + 5y - 9 = 0 \) (Here, \( a_1 = 8, b_1 = 5, c_1 = -9 \))
2. \( 3x + 2y - 4 = 0 \) (Here, \( a_2 = 3, b_2 = 2, c_2 = -4 \))

### Step 2: Set up the cross-multiplication formula
The formulas for \( x \) and \( y \) in terms of the coefficients are:

\[ x = \frac{b_1 c_2 - b_2 c_1}{a_1 b_2 - a_2 b_1} \]

\[ y = \frac{c_1 a_2 - c_2 a_1}{a_1 b_2 - a_2 b_1} \]

### Step 3: Substitute the values into the formulas
Now, we substitute the values of \( a_1, b_1, c_1, a_2, b_2, c_2 \):

For \( x \):

\[ x = \frac{5 \cdot (-4) - 2 \cdot (-9)}{8 \cdot 2 - 3 \cdot 5} \]

For \( y \):

\[ y = \frac{-9 \cdot 3 - (-4) \cdot 8}{8 \cdot 2 - 3 \cdot 5} \]

### Step 4: Simplify the expressions
Now, we simplify both expressions:

For \( x \):

\[ x = \frac{-20 + 18}{16 - 15} = \frac{-2}{1} = -2 \]

For \( y \):

\[ y = \frac{-27 + 32}{16 - 15} = \frac{5}{1} = 5 \]

### Step 5: State the solution
Thus, the solution to the system of equations is:

\[ x = -2, \quad y = 5 \]