Find the number of solutions a pair of linear equations `( 4 x - 9 y + 13 = 0 and 2 x + 3y - 13 = 0 )` have

Options

A. 1
B. 2
C. 10
D. Infinite

Answer

To find the number of solutions for the pair of linear equations given by:

1. \( 4x - 9y + 13 = 0 \)
2. \( 2x + 3y - 13 = 0 \)

we can follow these steps:

### Step 1: Rewrite the equations in standard form
The equations are already in standard form, which is \( Ax + By + C = 0 \).

### Step 2: Identify coefficients
From the first equation \( 4x - 9y + 13 = 0 \):
- \( a_1 = 4 \)
- \( b_1 = -9 \)
- \( c_1 = 13 \)

From the second equation \( 2x + 3y - 13 = 0 \):
- \( a_2 = 2 \)
- \( b_2 = 3 \)
- \( c_2 = -13 \)

### Step 3: Check the ratio of coefficients
We need to check the ratios \( \frac{a_1}{a_2} \) and \( \frac{b_1}{b_2} \):

\[ \frac{a_1}{a_2} = \frac{4}{2} = 2 \]

\[ \frac{b_1}{b_2} = \frac{-9}{3} = -3 \]

### Step 4: Compare the ratios
Since \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \) (i.e., \( 2 \neq -3 \)), we conclude that the lines represented by the equations are not parallel.

### Step 5: Determine the number of solutions
When the ratios of coefficients of \( x \) and \( y \) are not equal, it indicates that the two lines intersect at exactly one point. Therefore, the system of equations has exactly one solution.

### Conclusion
The number of solutions for the given pair of linear equations is **one**.

---