Parallel+and+Perpendicular+Lines

Parallel lines are lines which always have the same distance between them, and they will never cross.
Two non-vetical lines are parallel if they have the same slope, and two vetical lines are parallel for their slopes are undefined. For example:

What are perpendicular lines? How are the slopes of perpendicular lines related? Perpendicular lines or line segments intersect to form a right or 90 degree angle. The slopes of two non-vetical lines are perpendicular if the product of their slopes is -1, and a vertical line is perpendicular to a horizontal line. For example:

How can we find an equation of a parallel line when given a equation of a line and single point? To do that, first you need to know the point-slope form and how to find the slope in a standard equation of a line.

Example question: Write an equation of the line parallel to 3x+y-4=0 and through the point A(2, -5)

Step 1: When given an equation, find out the slope of the line based on the equation. Example: 3x+y-4=0, this equation follows the format of Ax+By-C=0 One way of finding out the slope is: -A/B, which is -3. since A=3 and B=1

Step 2: After you have the slope and one point, subsitute the known values into a point-slope form Example: y-y1=m(x-x1) y-(-5)=-3(x-2) y+5=-3x+6 3x+y-1=0 And there you go, the equation of the parallel line is 3x+y-1=0

How can we find an equation of a perpendicular line when given a equation of a line and single point? ​ Same as before, you need to know the point-slope form in order to do this correctly

Example question: State an equation of the lines perpendicular to the line 6x+3y-4=0 through the point A(3,5)

Step 1: Find the slope of the equation. Example: 6x+3y-4=0, this equation also follows the format of Ax+By-C=0 -A/B = -6/3, -6/3 = -2 now we know that the slope is -2

Step 2: For a perpendicular line, the slope of the equation of the perpendicular line would not be the same. So, to find out the slope of the perpendicular line, we need to find the negative reciprocal of the slope of the given equation, which is -2 To find the negative reciprocal, one way of doing this is to use this formular: M*(-2)=-1 m=1/2 Therefore, the slope of the perpendicular line is 1/2

Step 3: Sub in the know values into a point-slope form to find out the equation of the perpendicular line y-y1=m(x-x1) y-5=1/2(x-3) 2y-10=1(x-3) 2y-10=x-3 -x+2y-10+3=0 -x+2y-7=0 x-2y+7=0 Therefore, the equatioin of the perpendicular line is x-2y+7=0



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When the question doesn't give you a point, and instead, it wants you to have the parallel line or perpendicular line to have the same x or y intercept as another equation, we need an extra step in our solution.===== Solution: After you find the slope of the parallel or perpendicular line, you need to find the x or y intercept for the equation that asks you to have the same intercept. For example, the question asks for you to have the same y intercept as y=3x+5, as we all know, the y intercept is when x is zero, so y=3x+5, y=3(0)+5, y=5, therefore, the y intercept is 5 then, we need at least one point to sub into the point-slope form, so, we can use the y-intercept as a point(0, 6) Finally, we can sub in the point(0, 6) and the slope of the parallel or perpendicular line into a point-slope form