The point P(a,b) lies on the straight line `3x+2y=13` and the point `Q(b,a)` lies on the straight line `4x-y=5` , then the equation of the line PQ is

Options

A. x-y=5
B. `x+y=5`
C. `x+y=-5`
D. `x-y=-5`

Explanation

To find the equation of the line \( PQ \) given the points \( P(a, b) \) and \( Q(b, a) \) that lie on the lines \( 3x + 2y = 13 \) and \( 4x - y = 5 \) respectively, we can follow these steps:

### Step 1: Substitute point P into the first line equation
Since point \( P(a, b) \) lies on the line \( 3x + 2y = 13 \), we can substitute \( x = a \) and \( y = b \) into the equation:
\[ 3a + 2b = 13 \quad \text{(1)} \]

### Step 2: Substitute point Q into the second line equation
Point \( Q(b, a) \) lies on the line \( 4x - y = 5 \). Substituting \( x = b \) and \( y = a \) gives us:
\[ 4b - a = 5 \quad \text{(2)} \]

### Step 3: Solve the system of equations
We now have two equations:
1. \( 3a + 2b = 13 \)
2. \( 4b - a = 5 \)

We can rearrange equation (2) to express \( a \) in terms of \( b \):
\[ a = 4b - 5 \quad \text{(3)} \]

### Step 4: Substitute equation (3) into equation (1)
Now, substitute \( a \) from equation (3) into equation (1):
\[ 3(4b - 5) + 2b = 13 \]
Expanding this gives:
\[ 12b - 15 + 2b = 13 \]
Combining like terms results in:
\[ 14b - 15 = 13 \]
Adding 15 to both sides:
\[ 14b = 28 \]
Dividing by 14:
\[ b = 2 \]

### Step 5: Find the value of a
Now substitute \( b = 2 \) back into equation (3) to find \( a \):
\[ a = 4(2) - 5 = 8 - 5 = 3 \]

### Step 6: Identify points P and Q
Now we have the coordinates of points \( P \) and \( Q \):
- \( P(a, b) = (3, 2) \)
- \( Q(b, a) = (2, 3) \)

### Step 7: Find the equation of line PQ
Using the two points \( P(3, 2) \) and \( Q(2, 3) \), we can find the equation of line \( PQ \). The slope \( m \) of line \( PQ \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 2}{2 - 3} = \frac{1}{-1} = -1 \]

Using the point-slope form of the line equation \( y - y_1 = m(x - x_1) \):
\[ y - 2 = -1(x - 3) \]
Expanding this:
\[ y - 2 = -x + 3 \]
Rearranging gives:
\[ x + y = 5 \]

### Final Answer
The equation of the line \( PQ \) is:
\[ \boxed{x + y = 5} \]