How to use Algebra to find parallel and perpendicular lines. Show Parallel LinesHow do we know when two lines are parallel?
Their slopes are the same!
Example:Find the equation of the line that is:
The slope of y=2x+1 is: 2 The parallel line needs to have the same slope of 2. We can solve it using the "point-slope" equation of a line: y − y1 = 2(x − x1) And then put in the point (5,4): y − 4 = 2(x − 5) And that answer is OK, but let's also put it in y = mx + b form: y − 4 = 2x − 10 y = 2x − 6 Vertical LinesBut this does not work for vertical lines ... I explain why at the end. Not The Same LineBe careful! They may be the same line (but with a different equation), and so are not parallel. How do we know if they are really the same line? Check their y-intercepts (where they cross the y-axis) as well as their slope:
For y = 3x + 2: the slope is 3, and y-intercept is 2 For y − 2 = 3x: the slope is 3, and y-intercept is 2 In fact they are the same line and so are not parallel Perpendicular LinesTwo lines are Perpendicular when they meet at a right angle (90°). To find a perpendicular slope:
When one line has a slope of m, a perpendicular line has a slope of −1m In other words the negative reciprocal
Example:Find the equation of the line that is
The slope of y=−4x+10 is: −4 The negative reciprocal of that slope is: m = −1−4 = 14 So the perpendicular line will have a slope of 1/4: y − y1 = (1/4)(x − x1) And now put in the point (7,2): y − 2 = (1/4)(x − 7) And that answer is OK, but let's also put it in "y=mx+b" form: y − 2 = x/4 − 7/4 y = x/4 + 1/4 Quick Check of PerpendicularWhen we multiply a slope m by its perpendicular slope −1m we get simply −1. So to quickly check if two lines are perpendicular:
When we multiply their slopes, we get −1 Like this:
Are these two lines perpendicular?
When we multiply the two slopes we get: 2 × (−0.5) = −1 Yes, we got −1, so they are perpendicular. Vertical LinesThe previous methods work nicely except for a vertical line: In this case the gradient is undefined (as we cannot divide by 0): m = yA − yBxA − xB = 4 − 12 − 2 = 30 = undefined So just rely on the fact that:
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Erick L. A line passes through the point (0, 2) and has a slope of -1/2. What is the equation of the line? 3 Answers By Expert Tutors
Charles A. answered • 08/29/15 Charles A. Math, English, ESL, Economics, Finance
Typically you want to translate an equation into the "slope intercept" form y=mx+b, where y is your equation, also known as f(x), m is your slope (in this case -1/2) and b is your y-intercept (the y-value when x=0, given here by the point (0,2)). So all you have to do is write y=-1/2x+2 to find the equation of the line.
There are several forms for the equation of a line: the two most common are point-slope and slope-intercept. Point-slope form uses a point and the slope to get the equation. The equation for the line is y-y0=m(x-x0). The slope of the line is m, and a point on the line is (x0,y0). Slope-intercept form uses the slope and the y-intercept to get the equation. The equation for the line is y=mx+b. The slope of the of the line is m, and the y-intercept is b. (The y-intercept is the point where the line crosses the y-axis.) In the case of this problem, we have a point and a slope, so we want to use the point slope form. We can simply plug in the numbers to get y - 2 = -1/2(x - 0). This simplifies to y - 2 = -1/2 x, which is the equation of the line. Typically, however, people want the equation in slope-intercept form. This means that we need to solve for y, meaning we need to rewrite the equation so that it looks like y=something. In our case, if we add 2 to both sides of the equation, we are left with y = -1/2 x + 2, which is the slope intercept form of the equation for the line.
hint: the y.intercept is 2; the slope is given. You can answer it easily in the slope-intercept form or the point - slope form. But just for fun, let's track down the x intercept. Then you might want to write the equation in Standard Form. The slope is rise/run = -1/2, so as the line moves to the right 2 units, it moves down 1 unit. Starting from (0,2) and moving to the right 2 units, the line goes down 1 unit to the point (2,1). Again moving on by going 2 units right and down one unit, the line intersects the x.axis at (4,0). For standard form, Ax + By = C, find C: C = the x.intercept times the y.intercept When y = 0, at the x intercept, x = 4, and A would have to be 2 so that A(4) + B(0) = 8. When x = 0, at the y intercept, y = 2, and B would have to be 4 so that A(0) + B(2) = 8. So A = 2 and B = 4 for your standard form. |