The equation of a straight line passing through the point of intersection of x…

Step 1: Find the intersection point of the two given lines.

Solve x − y + 1 = 0 and 3x + y − 5 = 0.

From x − y + 1 = 0, y = x + 1. Substitute into 3x + y − 5 = 0: 3x + (x + 1) − 5 = 0 ⇒ 4x − 4 = 0 ⇒ x = 1, y = 2.

Intersection point: (1, 2).

Step 2: Find slopes of the given lines.

• For x − y + 1 = 0 → y = x + 1, slope = 1.

• For 3x + y − 5 = 0 → y = 5 − 3x, slope = −3.

Step 3: Determine lines through (1,2) that are perpendicular to each given line.

1. Perpendicular to the first line (slope 1): perpendicular slope = −1. Equation through (1,2): y − 2 = −1(x − 1) ⇒ x + y − 3 = 0.

2. Perpendicular to the second line (slope −3): perpendicular slope = 1/3. Equation through (1,2): y − 2 = (1/3)(x − 1) ⇒ x − 3y + 5 = 0.

Step 4: Compare with the provided choices and conclude.

The line x − 3y + 5 = 0 is the line perpendicular to 3x + y − 5 = 0 that passes through the intersection point (1,2), so it is the correct answer.