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Find the point on the line $y=2 x+3$ that is equidistant from the points $(-5,6)$ and $(0,0)$.

(25 / 14,46 / 7)

Algebra

Chapter 1

Functions and their Applications

Section 1

The Line

Functions

Campbell University

Oregon State University

Baylor 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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04:08

Find the point on the $y$ …

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The point $\left(x_{0}, y_…

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Find the equation of the s…

01:02

Find a point that is equid…

06:22

Find the point equidistant…

Find the point on the $x$ …

for this problem. We've been given a line. Why equals two X plus three. And then we've been given two points and we want to find a point on this line that is equally distant from the two points given. Now, these two points that were given do not actually fall on the line. But we want the point on the line. That's the same distance from both of these points. Okay, so since we're looking at the distance between them, we're going to be using the distance formula. So as a review, the way we would put together a distance formula, the distance is the square root of two things. We want to find the difference between my exes squared. So between my two points, I'm gonna find the district difference between the X coordinates, square it, and I'm gonna add to that the difference between my Y coordinates and I'm going to square that Ah, this does stem from the Pythagorean Theorem s. Oh, this is our distance formula, and we're just gonna have to use it twice. We want to know how far the point is to the first point and how far the point on the line is to the second point. So what is the point on the line that we're comparing them? Thio? Well, it's a point, and we'll just call it X y. Now, I would like to have this in terms of a single variable. And because I'm given the equation of the line, I know that why equals two X plus three. So I can use that as my point if I know what the exes. I confined the y by going to X plus three. Um, and I'll know that that's gonna be on my line. So let's take a look. That green point to the first point. Negative. 56 That distance is the square root of the difference in my exes. So X minus negative five four x plus five squared the difference. In my wise, I have two x plus three minus six squared. Right. And just to simplify this a little bit this year is going to be This is two X minus three squared. Okay, Now that distance has to equal the distance to the second point, the 0.0.0 Okay, so the difference of my exes X minus zero squared, plus the difference in the wise to X plus three minus zero squared. Well, the good thing is, with those zeros, yeah, we can just ignore them or just subtracting zero. Okay, next line. First of all, I'm going to square both sides That gets rid of the square root side so I can I can square both sides and that goes away. So let's expand this out. On the left hand side. I have X squared plus 10 x plus 25 and then I can square the second piece plus four X squared, minus 12 x plus nine. Now let's look at the right hand side. That first term is just X squared. The second term is four x squared plus 12 x plus nine. Okay, well, the nice thing is, I have some things to cancel. I haven't X squared on both sides as well as a four x squared and a nine. What do we have left? Well, on the left hand side, I have 25 minus two x on the right hand side. I have 12 x added to x to both sides and divide and I get X equals 25/14. Not a nice number, but it's a but that is the answer. X X is a nice fraction 25/14. Now let's find the wise. I'm going to go back toe what I know about my line. Why equals two X plus three. So in this case, why equals two times 25/14 plus three? I can cancel that, too. To that first piece is now 25/7. If I want a common denominator, that's gonna be 21/7, which equals 46/7. So that's my y, coordinate. So my my answer, the point that is equidistant from the two given points is the 20.25 14th 46 7th.

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