Lesson Linear equations and slopes: Special Cases Part I

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This Lesson (Linear equations and slopes: Special Cases Part I) was created by by richwmiller(17219)

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About richwmiller: I like math.

I would like to touch on some frequently occurring problems regarding linear equations and slopes.
There are two special linear equations.
{{{y=4}}} (or any number) and {{{x=4}}} (or any number)
They are perpendicular.

A) {{{y=4}}}
Y never changes. X can be anything but y is always 4.
{{{y=4}}} is a horizontal line across the page.
Following the slope intercept formula equation {{{y=mx+b}}}
{{{m =0}}}
Thus there is no x since {{{0*x=0}}} and x does not appear at all.
To understand what is happening remember that the slope is the rise over the run {{{rise/run}}}
The rise is the change in y and the run is the change in x.
The rise y goes up and down and the run x goes across the page.
So if y never changes then the change in y is zero. So the fraction becomes {{{0/run}}} which is zero.
So the slope m of {{{y=4}}} is 0.
It has no x intercept because y never equals zero and the line never crosses the x axis.

B) {{{x=4}}}
x never changes.  y can be anything but x is always 4.
The other special linear equation is {{{x=4}}} (or any number).
Remember that {{{y=4}}} is horizontal and is perpendicular to {{{x=4}}}
So {{{x=4}}} is vertical (up and down).
What is special about that?
We discussed the slope of {{{y=4}}} and found it is zero because the rise is zero. Here the run is zero and the rise can be anything. But what happens to the fraction when the run equals zero. The denominator is zero. We cannot divide by zero. That operation is undefined. So we have {{{rise/zero}}}. The slope of {{{x=4}}} is {{{undefined}}}. X never changes. So the change in y over the change in x becomes rise over zero. {{{rise/0}}}
There is no y intercept because x never equals zero and the line never crosses the y axis.

There is something else special about {{{x=4}}}. {{{It_is_ not_ a_ function}}}. A function must have one to one relationship from x to y. But for this x, there are many y's.

Practice problems:

Problem 1
Need help solving 4x=8 and 5y=15 using substitution or elimination, please!
Here is a case of y=4 and x=4 together.
Answer:
Here they work out to be y=3 and x=2
These two equations are perpendicular. You don't solve them together.
Solve them individually because one has no x and the other has no y.

Problem 2
I was given this problem to solve:
f(x) = 12/7
Evaluate for f(6).
I cannot see how this is a function, since there is only one variable on the left side of the equation.  The answer I submitted was that there was no real solution and I was graded as wrong.  Can  you tell me how to solve this problem? 
Answer :
No matter what x is f(x)=12/7
So f(6) is still 12/7
It is a function. y can have many x's (think zeroes or roots) but x can have only one y. So x=12/7 is not a function.

Problem 3
What is the y coordinate of the y intercept of the line through the points (2,5) and (4,5)
Answer:
An equation for these points is y=5
When x=0, y=5.