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Determine an equation for the line (a) parallel (b) perpendicular to $2 x-5 y=9$ and passing through the point $(-2,-4)$.

(a) $2 x-5 y=16$(b) $5 x+2 y=-18$

Algebra

Chapter 1

Functions and their Applications

Section 1

The Line

Functions

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University of Michigan - Ann Arbor

Idaho State University

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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for this problem, we're gonna be finding two different equations of lines, one that's parallel and one that's perpendicular to a given line. Now, the line we're gonna be examining is two x minus five Y equals nine. And in both of these cases, we're also going to go through a point that both of these lines they're going to go through the point negative to negative four. Okay, so let's take a step back. What do we mean by parallel and perpendicular? Parallel lines, our lines there side by side, but never touch. That means that they are rising or falling in exactly the same rate. That rate at which you rise and fall is the slope, so parallel lines have equal slopes. So if I find the slope of the line that I'm comparing it to my parallel line will have an identical slope. What about perpendicular lines? Well, perpendicular lines are lines that meet at a right angle. Now those obviously do not have the same slopes because they cross each other but their slopes are related. Perpendicular lines have slopes that are negative, reciprocal of each other, so negative, reciprocal. In other words, if one has a slope of one half. The other will have the negative, Reciprocal or negative to If one slope is 373 other is the negative reciprocal negative. Seven thirds, right? So how this race that because those numbers were just examples, not part of our actual problem. So in both of these cases, we're gonna need to find the slope of our original line that we're comparing them to. So let's put this into slope intercept form. Why equals M X Plus B? If we can do that, then this m is our slope. We confined the other lines that we need. Okay, So let's solve our original equation for why everything is not why I'm gonna put over to the right hand side. So I have negative five y equals nine. Let's put the extreme first negative two x plus nine. And then I'm going to divide everything by negative five. That gives me why equals 2/5 X minus 9/5. Okay, so my slope for my original line is 2/5 for my parallel line. That slope will also be 2/5 for the perpendicular line. I have a slope. This the negative reciprocal or negative five halves. Okay, so let's find our lines parallel. First we have a point and a slope. Remember both of these trying to go through the point. Negative to negative four. I have point in slope, so I'm going to use the point slope form of the line. Why? Minus y one equals m times X minus X one where X one y one is my point. M is my slope. So for the parallel lines, I have Why minus negative four, which is why plus four. My slope is the same as the other line, which is 2/5 X minus negative two or X plus two. And I can make this look a little nicer. Let me get rid of my parentheses. That gives me to Fifth X plus 4/5. And then I'm going to subtract four from both sides. When I subtract four, the four on the left hand side is gone and I subtract four and I'm gonna give it a common denominator of 20/5. So that gives me a final form. Why equals to 50 X minus 16 5th. Okay, so that's my parallel life. What about perpendicular? And I'm just gonna abbreviate that well again, I'm gonna use the same thing. Why? Minus y one equals m times X minus X one same format and part of it's gonna look identical because my ex and my why are the same. So those pieces don't change. The only thing is different is the slope. And in this case, my slope is now negative. Five hats again. I'm gonna get rid of my parentheses. That's negative. Five Have X minus five. And this gives me why equals negative five halves X Subtracting four from both sides gives me a minus nine. So those are my two equations. One that's parallel one that's perpendicular.

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