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The area of a triangle formed by a line and the two axes is 40 and the slope of the line is $-5 .$ Find an equation for the line. (Two possible answers.)

$y=-5 x \pm 20$

Algebra

Chapter 1

Functions and their Applications

Section 1

The Line

Functions

Harvey Mudd College

Baylor University

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. I want to find two possible lines that match some given criteria. What I'm told is, the slope of my line is going to be negative. Five. And I want the light to cut off a triangle with the axes tohave an area of 40 square units. So let's look at that area first. What? Let's see if we can visualize what that looks like if I just sketch in a really rough sketch here. If I have a slope of negative five, that means I'm gonna be coming down fairly steeply. Negative five means I have a downward line. Five is a pretty steep slope, so that's going to cut off an area of a triangle here. I know what that area to be 40 but the director to say we should expect to possible lines. So here's our other option. I could also I have to have the same slope, so these are gonna be parallel lines. They'd be more parallel if I was actually using pencil and paper and a ruler. But we're gonna pretend those lines of parallel, and it also could cut off a area down there in the third quadrant that Blue Triangle. So we want those areas to be 40. Well, let's think for a moment. Let's just give us, uh understand marks. Um, things on here area of a triangle equals one half the base times the height. Now, I'm just gonna be really clear here. We've been using B a lot in this lesson for the y intercept for this problem be is base is the base of our triangle. Okay, so I just wanna be very clear. I'm not gonna use be for why Intercept B is the base h is my height. Okay, so let's look at that green. Uh, let's look at the green triangle. Just toe. Be able to do some reference stuff here. I need to know the base. Well, the base. Well, if I know my why intercept that will tell me how big my bases. And so let's call it since it's be if I want this distance here to be be my ex intercept will be be zero head. What about my height? Well, if I want my height to be h, then my y intercept will be zero h okay. And I want that area thio equal 40 so If I multiply both sides by two base times, height here is gonna have to equal 80. Well, this is good so far, but I need some more information. I've got two variables here. Now is when I can come back and look at my slope. Remember that how we find Slope Slope is our rise over one or change And why over our change in X. So when we go to find Slope Weaken right, it is. Why one minus y two over X one minus x two. That's our slope. Well, in this case, I know that my slope is negative five. And that's gonna equal. And I'm going to take I'm just gonna circle this in red here. I'm gonna let that be my first point b zero will be my second point. So when I go to find the slope, I take the Why coordinate? It's the first point minus the second point, and the X coordinates first point minus second point. So that gives me H over negative B. So if I do some cross multiplication here that gives me five b equals H, I can plug that back into my equation, so I could take this be times eight equals 80. And I could say that that's b times five b equals 80. So now I can solve this. Five b squared equals 80 or B squared equals 16. So be is going to be either plus or minus four. So the plus four will give me three X coordinate for the green triangle. Them negative four will be for the blue triangle that I have drawn over there. So that's my base. Now. What about my height? Okay, I'm just gonna change my color, actually. Can't keep green here. I know that five times b equals h. So in this case, five times either plus or minus four will give me H So H is going to be plus or minus 20. Again, the positive will correspond with the green triangle, the negative of correspond with the blue triangle. Okay, so I have the basin heights of my triangles. Now, what I really want is the equation of the line. Okay, so let's go right thumb for the first one for the Green Triangle. I know that my slope is negative five. And I know my y intercept, which is going to be at the 50.0 20. So I'll have y equals m X plus B negative five X plus 20. If I look at my blue triangle that why intercept is going to be down at zero with negative value of the H same slope. So it's gonna be negative five x minus 20. So if I just use the plus remind us here, that gives me both of the possible, uh, lines to satisfy my given conditions.

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