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(a) Plot the line $4 x+6 y+12=0$. Find the area of the triangle formed by the line, the $x$ -axis and the $y$ -axis. (b) Repeat for $A x+B y+C=0$ for $A$$B, C$ positive.

(a) 3(b) $\frac{C^{2}}{2|A B|}$

Algebra

Chapter 1

Functions and their Applications

Section 1

The Line

Functions

Campbell University

McMaster University

University of Michigan - Ann Arbor

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 going to look at the equation for X Plus six y plus 12 equals zero. We're gonna find some information about this line. Then we're going to finish this problem by looking at a more generic case. So first, here's our line. We want to plot it. So in order to plot this, let's put this into slope intercept form. Remember, Slope intercept four means we're solving for why so everything. That's not why I'm gonna put to the right hand side. So that becomes a negative four X minus 12. And I'm going to divide both sides by six. Simplifying all my fractions, I get why equals negative two thirds X minus two. So there's my equation. If I'm gonna go plot this, remember this slope intercept form. So my why intercept is negative two. So I can plot that point on my grid and I have a negative slowly to my line would be going down. My rise overrun is to over three. So I have a negative fries, which is a drop of two, and I've run three, dropped to over three, and I could also come up 123 and do the same thing. So here's my line. Hey, so there's my line. Now, the next question I have here is what's the area of the triangle formed by the axes and this line? So you can see I've got a nice little triangle here and I want to find the area. Well, little review from geometry. The area of a triangle is one half base times height. Now, we've been using be in this lesson to talk about our Y intercept. So what I'm going to do here is I'm actually gonna write this out. I know we usually just do b and H, but I'm gonna dio base and height just because I don't want there to be any confusion about these variables. Okay, So what is the base and height of this triangle? Well, if I imagine it, um, I'm just gonna kinda looking here. I'm gonna let the base be the y axis. I'm kind of Let's sit on that axis. I know that I'm two units on the Y axis because my wife intercept was too. And I'm not going to say it's negative too, because this is an area that's a distance along the side of a triangle. You can't have a side of a triangle equal to a negative number. So even though I intercepted that negative to the length of that base is two units. So what's my height? Well, for my height, I need to know what my ex intercept is now. I could do a pretty good job of eyeballing it from my picture here, but just in case you don't know for sure if the picture is not very clear or if you don't wanna actually plot it, you're just trying to do this algebraic Lee. We find our X intercept by letting why equals zero that gives us four X plus 12 equals zero. And if you solve for X, you get X equals negative three. So, again, my my ex intercept is negative three. But that distance that height has to be positive. I can't have a negative length on a triangle. And when I do this out, I get an area of three square units. So what if I made this a little more generic? Here's my generic equation. Would have a X plus B y plus C equals zero. What is the generic? Um, equation. Actually, I don't even need the equation. Really? What I really need to know is the area. That's that's That's my goal. So what I needed to find my area was I used my Why intercept That gave me my base is my ex intercept to give me the height. So let's find them by. Why intercept the Y intercept means that my ex of coordinate zero. So that just means I have b y plus C equals zero. And if I saw for why, means I'm gonna be subtracting C and dividing by B so my Y intercept is negative C over B, which means the base will be C over B at what about my ex intercept? On that case, my Y value is zero. So this becomes a X plus C equals zero and solving for X. I get negative, see over a So when I want the area here one half the base, that's my why I get that from my white intercept. She'll be multiplying that by C over b my height I get from my ex intercept that will be see over a. And I could also write this and C squared over to a B And since we're told that A, B and C are all positive, I don't have to worry about the signs here. I know this is going to give me a positive number, so this will be the area of that triangle.

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