If the line `x/a+y/b=1` passes through the points a `(2,- 3)` and `(4, - 5),` then `(a, b) =`

Options

A. `(1,1)`
B. `(-1,1)`
C. `(1,-1)`
D. `(-1,-1)`

Explanation

To solve the problem, we need to find the values of \( a \) and \( b \) such that the line given by the equation

\[ \frac{x}{a} + \frac{y}{b} = 1 \]

passes through the points \( (2, -3) \) and \( (4, -5) \).

### Step 1: Substitute the first point into the line equation
We start by substituting the first point \( (2, -3) \) into the line equation:

\[ \frac{2}{a} + \frac{-3}{b} = 1 \]

This simplifies to:

\[ \frac{2}{a} - \frac{3}{b} = 1 \quad \text{(Equation 1)} \]

### Step 2: Substitute the second point into the line equation
Next, we substitute the second point \( (4, -5) \) into the line equation:

\[ \frac{4}{a} + \frac{-5}{b} = 1 \]

This simplifies to:

\[ \frac{4}{a} - \frac{5}{b} = 1 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
Now we have a system of two equations:

1. \( \frac{2}{a} - \frac{3}{b} = 1 \)
2. \( \frac{4}{a} - \frac{5}{b} = 1 \)

Let's multiply Equation 1 by 2 to make the coefficients of \( \frac{1}{a} \) in both equations the same:

\[ \frac{4}{a} - \frac{6}{b} = 2 \quad \text{(Equation 3)} \]

### Step 4: Subtract Equation 2 from Equation 3
Now, we subtract Equation 2 from Equation 3:

\[ \left( \frac{4}{a} - \frac{6}{b} \right) - \left( \frac{4}{a} - \frac{5}{b} \right) = 2 - 1 \]

This simplifies to:

\[ -\frac{6}{b} + \frac{5}{b} = 1 \]

\[ -\frac{1}{b} = 1 \]

### Step 5: Solve for \( b \)
From the equation \( -\frac{1}{b} = 1 \), we can solve for \( b \):

\[ b = -1 \]

### Step 6: Substitute \( b \) back into Equation 1 to find \( a \)
Now that we have \( b = -1 \), we can substitute this value back into Equation 1:

\[ \frac{2}{a} - \frac{3}{-1} = 1 \]

This simplifies to:

\[ \frac{2}{a} + 3 = 1 \]

Subtracting 3 from both sides gives:

\[ \frac{2}{a} = 1 - 3 \]

\[ \frac{2}{a} = -2 \]

### Step 7: Solve for \( a \)
Now, we can solve for \( a \):

\[ a = -1 \]

### Final Result
Thus, we have found:

\[ (a, b) = (-1, -1) \]