Equations for Intersecting Lines

The point of intersection on a graph represents the coordinates, or conditions, under which two equations are both true.

Many of the characteristics of intersecting lines, such as their slopes and intercepts, can be calculated by analyzing the lines separately because these characteristics are not affected by each line's intersection with the other line. In contrast, finding the point of intersection requires analyzing both lines.

Slope

Straight line equations follow the format y = mx + b, where m and b are constants (i.e., numbers as opposed to variables). This format is called the slope-intercept equation. The letter m represents the slope of the line. So, for example, if the equation of a line is y = 3x + 5, the slope of the line would be 3.

Y-Intercept

In the equation y = mx + b, the letter b represents the y-coordinate of the y-intercept of the line. The y-intercept is the point at which a line crosses the y-axis. Any point on a graph has both an x-coordinate and a y-coordinate. The x-coordinate at any point on the y-axis is zero, and so finding the y-coordinate of the y-intercept allows you to figure out exactly where the y-intercept is located. So, for example, if the equation of a line is y = 3x + 5, the y-intercept would be (0, 5).

X-Intercept

The equation y = mx + b does not directly tell you the coordinates of the x-intercept of the line but it does provide enough information to solve for the x-intercept. The x-intercept is the point at which a line crosses the x-axis. The y-coordinate at any point on the x-axis is zero. Because the y-coordinate at the x-intercept is zero, set y equal to zero in the equation y = mx + b and then solve for x. So, for example, if the equation of a line is y = 3x + 5, you would have 0 = 3x + 5. Solving for x, you would get -5 = 3x, so x = -5/3. The x-intercept is thus (-5/3, 0).

Point of Intersection

When two lines intersect, their x- and y-coordinates are equal at the point of intersection. The lines' equations can therefore be set equal to each other in order to solve for the point of intersection. So, for example, if the equations of two intersecting lines are y = 2x + 1 and y = -x + 4, you would have 2x + 1 = -x + 4. Solving for x, you would get x = 1. The second step in solving for the point of intersection is to plug its x-value into either equation and solve for y (e.g., y = 2(1) + 1 = 3 or y = -1 + 4 = 3).

References

• Math Open Reference; Intersecting Straight Lines; John Page; 2009
• AlgebraLAB; Intercepts of Lines; S. Taylor; 2011