Solve, using cross - multiplication :
x - y + 2 = 0
7x + 9y = 130

Answer

To solve the simultaneous equations using cross-multiplication, we have the following equations:

1. \( x - y + 2 = 0 \)
2. \( 7x + 9y = 130 \)

### Step 1: Rewrite the equations in standard form
The first equation can be rewritten as:
\[ x - y = -2 \]
This can be expressed in the form \( ax + by + c = 0 \):
\[ 1x - 1y + 2 = 0 \]

The second equation can be rearranged as:
\[ 7x + 9y - 130 = 0 \]

### Step 2: Identify coefficients
From the equations, we can identify the coefficients:
- For the first equation \( a_1 = 1, b_1 = -1, c_1 = 2 \)
- For the second equation \( a_2 = 7, b_2 = 9, c_2 = -130 \)

### Step 3: Set up the cross-multiplication formula
The cross-multiplication formula for solving the equations is given by:
\[ \frac{x}{b_1 c_2 - b_2 c_1} = \frac{y}{c_1 a_2 - c_2 a_1} = \frac{1}{a_1 b_2 - a_2 b_1} \]

### Step 4: Calculate the denominators
Now we will calculate each part of the formula.

1. Calculate \( b_1 c_2 - b_2 c_1 \):
\[ b_1 c_2 - b_2 c_1 = (-1)(-130) - (9)(2) = 130 - 18 = 112 \]

2. Calculate \( c_1 a_2 - c_2 a_1 \):
\[ c_1 a_2 - c_2 a_1 = (2)(7) - (-130)(1) = 14 + 130 = 144 \]

3. Calculate \( a_1 b_2 - a_2 b_1 \):
\[ a_1 b_2 - a_2 b_1 = (1)(9) - (7)(-1) = 9 + 7 = 16 \]

### Step 5: Substitute into the formula
Now we substitute these values into the cross-multiplication formula:
\[ \frac{x}{112} = \frac{y}{144} = \frac{1}{16} \]

### Step 6: Solve for \( x \) and \( y \)
From the first part:
\[ \frac{x}{112} = \frac{1}{16} \implies x = \frac{112}{16} = 7 \]

From the second part:
\[ \frac{y}{144} = \frac{1}{16} \implies y = \frac{144}{16} = 9 \]

### Final Solution
Thus, the solution to the simultaneous equations is:
\[ x = 7, \quad y = 9 \]