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Show if $a b \neq 0,$ then the line with intercepts $(a, 0)$ and $(0, b)$ has the equa$\operatorname{tin} \frac{x}{a}+\frac{y}{b}=1$.

Algebra

Chapter 1

Functions and their Applications

Section 1

The Line

Functions

Campbell University

Harvey Mudd College

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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01:42

Using Intercepts Show that…

01:13

Show that the equation of …

03:39

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06:43

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01:26

Prove that if a line $L$ h…

for this problem. We're going to look at a generic line that has two intercepts. A zero would be the X intercept. I zero b would be the Y intercept. Now we're gonna make one caveat. We do not want this to go through the origin, so A and B are not going to be zero. Okay, maybe you're gonna be any number except zero. Now, let's take a look at this. I'm going to draw this as if a and B a positive, but I'm not gonna make any assumptions about the side. So what I'm gonna do here, even though the picture shows them is positive, the math will work for any A or B positive or negative, as long as it's not zero. So if I plot in my, uh, intercepts there is a zero, and here is zero B and I draw that line. Okay, there's my line. Goes through zero b. It goes through a zero. What is the equation of the line that goes through those points? Well, the good thing is, is we have some points, and I specifically I have the y intercept. So if I can find the slope, we could use the slope intercept form. Well, we confined the slope. Recall slope is the change in Why are rise over the change in X, which is our run. So if I have two points, I could just take the difference of the why variable or the white coordinates and subtract the difference between the X coordinates. So let's do that here. I'm going to start with my why intercept. This will be my first point. So I have B minus zero. The difference of my why coordinates over zero minus a the difference of my ex coordinates. So that gives me negative. Be over a for a slope, Put that in a slope Intercept form. I have y equals negative b over a X, and we're told that our Y intercept is B so let's just do a little manipulation here. I'm going to move the X, so it's on the same side as the UAE, so that would give me a positive. Be over eight times X plus. Why equals B if you look at the problem, we want to solve this so that it equals one. So I'm going to divide every term by B all the way across going to be in the denominator my first term, that's going to cancel. So I just have X over a. Then I have y over B, and that equals one again. I made no assumptions about whether A and B re positive or negative through here, so this will work for any intercepts that were given.

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