Find the equation of the line that passes through the point (2, -3) and makes an angle of -45° with the positive X-axis. Use the slope-angle relation m = tan θ and write the final equation in simplest standard form.

Introduction / Context: This coordinate geometry question tests how to form a line equation from a point and an angle of inclination. The angle that a line makes with the positive X-axis determines its slope through the relation m = tan θ. Once the slope is known, the point-slope form y - y1 = m(x - x1) gives the line equation directly. The main care points are handling the negative angle (which creates a negative slope) and simplifying to a clean standard form.

Given Data / Assumptions:

• Line passes through (x1, y1) = (2, -3) • Angle with positive X-axis: θ = -45° • Required: equation of the line

Concept / Approach: Use slope-angle relation: m = tan θ. Then use point-slope form: y - y1 = m(x - x1). Finally simplify into standard form (like x + y = constant) if possible. Since tan(-45°) = -1, the line has slope -1, which is a common and easy slope to simplify.

Step-by-Step Solution: 1) Compute slope from the given angle: m = tan(-45°) = -1 2) Use point-slope form with (2, -3): y - (-3) = -1(x - 2) 3) Simplify: y + 3 = -x + 2 4) Bring x to the left and constants to the right: x + y + 3 = 2 5) Simplify the constant: x + y = -1

Verification / Alternative check: Substitute the given point (2, -3) into x + y = -1: 2 + (-3) = -1, which matches the right side, so the line passes through the point. Also, the slope of x + y = -1 is -1 (since y = -x - 1), matching tan(-45°) = -1. Both conditions are satisfied.

Why Other Options Are Wrong: • x + y = -5 and x - y forms correspond to different slopes and do not satisfy both the slope and point condition. • y = x - 5 has slope +1, which would correspond to +45°, not -45°.

Common Pitfalls: • Using tan(-45°) as +1 instead of -1. • Misplacing the sign in y - y1 = m(x - x1), especially since y1 is negative here. • Stopping at y + 3 = -x + 2 without simplifying to a clean final form.

Final Answer: x + y = -1