How do you graph \frac { 3y - 5x } { 2} = \frac { y x } { 2} + 4?

1 Answer
Aug 10, 2017

First, separate the x's and y's. You want y as a function of x.

Explanation:

...there's more than one way to do this, here's how I did it.

Multiply both sides of the initial eq. by 2:

3y - 5x = yx + 8

Subtract yx from both sides:

3y - 5x - yx = 8

Add 5x to both sides:

3y - yx = 8 + 5x

Now, on the left side, factor out y:

y(3 - x) = 8 + 5x

Divide both sides by (3 - x):

y = (8+5x)/(3 - x)

So now you have y expressed as a function of x.

As to graphing it, first find y when x = 0:

y = 8/3

and then, note that when x = 3, the denominator is zero. You can't divide by zero, so you can't graph this point. But imagine when x is just a TINY bit less than 3. Then 3 - x is very close to zero.

A number divided by a tiny, tiny number is very large. So you know that the graph runs off to positive infinity as x approaches 3 from the left.

Now, imagine x is just a tiny bit greater than 3. Now 3 - x is a tiny, tiny NEGATIVE number. So you then know that the graph climbs up from NEGATIVE infinity as x proceeds from points to the right of x = 3.

Now: this is algebra, not calculus. Calculus gives you some tools to better graph this function, but I can't use them here. So then calculate y for points where x = 1, 2, 4, 5, and maybe several more. and also -1, -2, -3, etc.

And then connect the dots!

You should get a form of a hyperbola. You can check the result:

https://www.desmos.com/calculator

(paste in (8+5x)/(3-x))