The equation of a line is typically written as y equals m x plus b, where m represents the slope and b is the y-intercept. The slope measures how steep the line is, calculated as the ratio of the vertical change to the horizontal change. In this example, our line has a slope of 1 and a y-intercept of 1, giving us the equation y equals x plus 1. The y-intercept is the point where the line crosses the y-axis.
Parallel lines have the same slope but different y-intercepts. If one line has the equation y equals m x plus b-one, then any line parallel to it will have the equation y equals m x plus b-two, where b-two is different from b-one. In this example, we have two parallel lines with the same slope of 1. The blue line has a y-intercept of 1, giving us the equation y equals x plus 1. The red line has a y-intercept of negative 2, giving us the equation y equals x minus 2. Notice how both lines rise at the same rate, but they never intersect because they have different y-intercepts.
Perpendicular lines have slopes that are negative reciprocals of each other. If one line has a slope of m-one, then any line perpendicular to it will have a slope of m-two, where m-one times m-two equals negative one. In this example, the blue line has a slope of 2, and its equation is y equals 2x minus 1. The red line has a slope of negative one-half, and its equation is y equals negative one-half x plus 2. Notice that 2 times negative one-half equals negative one, confirming that these lines are perpendicular. The lines intersect at the point (1.2, 1.4), forming a right angle. If a line is horizontal with a slope of zero, then its perpendicular line will be vertical with an undefined slope.
Let's work through an example of finding parallel and perpendicular lines. Given the line y equals 3x minus 2 and the point (2, 1), we want to find the equations of lines that pass through this point and are either parallel or perpendicular to the given line. For a parallel line, we use the same slope, which is 3. Then we substitute the point (2, 1) into the equation y equals 3x plus b to find the y-intercept. We get 1 equals 3 times 2 plus b, which gives us b equals negative 5. Therefore, the parallel line has the equation y equals 3x minus 5. For a perpendicular line, we use the negative reciprocal of the slope, which is negative one-third. Substituting the point (2, 1), we get 1 equals negative one-third times 2 plus b, which gives us b equals five-thirds. Therefore, the perpendicular line has the equation y equals negative one-third x plus five-thirds. On the graph, we can see the original line in blue, the parallel line in green, and the perpendicular line in red, all passing through the given point in purple.
To summarize what we've learned about parallel and perpendicular lines: First, a line's equation is y equals m x plus b, where m represents the slope and b is the y-intercept. Parallel lines have the same slope but different y-intercepts. This means they maintain the same angle with the x-axis and never intersect. Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product equals negative one. These lines intersect at a right angle. To find a parallel line through a specific point, use the same slope as the original line and substitute the point's coordinates to solve for the y-intercept. To find a perpendicular line through a point, use the negative reciprocal of the original slope and then solve for the y-intercept. These concepts are fundamental in coordinate geometry and have applications in fields like physics, engineering, and computer graphics.