Intersection of Two Lines in the Coordinate Plane: A Detailed Guide

In the coordinate plane, the intersection of two lines can be determined using their slopes and intercepts. Let's explore how to find the point of intersection for two lines given their slopes and intercepts.

Given Data

Line A has a slope of -1 and an x-intercept of 1. Line B has a slope of 5 and a y-intercept of -5. We need to find the coordinates of the point of intersection, denoted as (a, b), and the sum a b.

Equations of the Lines

Let's derive the equations of the lines using the given information.

Line A

Line A has a slope of -1 and an x-intercept of (1, 0). The general form of the line's equation with a given slope is:

Substituting the point (1, 0) into the equation, we get:

Solving for c, we find that c 1. Therefore, the equation of Line A is:

Line B

Line B has a slope of 5 and a y-intercept of -5. The general form of the line's equation with a given slope and y-intercept is:

Substituting the given y-intercept, we get:

Finding the Intersection Point

To find the intersection point (a, b), we set the equations of the two lines equal to each other:

Solving for x, we get:

Substituting x 1 back into either of the equations, we find:

Therefore, the intersection point (a, b) is (1, 0).

Conclusion

The sum of the coordinates a b at the intersection point is:

Thus, the sum a b is 1.