Expert: Scott A Wilson - 9/1/2011

Question
Hi,

I got a response saying you couldn't locate the question I linked to my email:

I went ahead and cut/pasted it below. The very top has your name as the expert so I assumed it was a question you answered.  

I did get your response about solving

Ax + By = D
Bx + cy = E

and your sample points: (2,2), (3,6),(4,10)  solving the equations I get y=4 and x=-6, but I'm still a little confused on what I do next with the x=-6 and y=4 to find the equation?  I am now interested in finding out more about both your response to my original email and the below response.  Thank you for taking the time to answer my questions.  


Expert: Scott A Wilson - 6/11/2009

Question
QUESTION: Hello,
I'm in the 6th grade and I cannot find the rule from the follwing table.

x=1, 5, 9
y=5, 1, -3
I cannot find the pattern. What am I missing?


ANSWER: What you are missing is that you need to look at the differences between each consecutive element.
For the x values, 5 - 1 = 4, and 9 - 5 = 4.
For the y values , 1 - 5 = -4, and -3 - 1 = -4.

Since each time x increases, y decreases by the same amount,
this says to look at is x+y.

For the first terms, 1 and 5, the sum is 1 + 5 = 6.
For the second terms, 5 and 1, the sum is 5 + 1 = 6.
For the third terms, 9 and -3, the sum is 9 + -3 = 6.

That says that we have the equation x + y = 6.
A formal equation usually solves for y in terms of x,
so subtract x from both sides and get y = 6 - x.

That would be what you need to know.
You have two choices: stop here or read on for even more info.

=====================================================================

Now that was the special case where x and y vary by the same amount.

Suppose we had X = 1, 5, 9 and y = 10, 8, 6.
Note that here, every time x increases by 4, y decreases by 2.
This says multiply the y value by 2 and add to the x.

The first numbers are 1 and 10, so 1 + 2*10 = 1 + 20 = 21.
The second numbers are 5 and 8, so 5 + 2*8 = 5 + 16 = 21.
The third numbers are 9 and 6, so 9 + 2*6 = 9 + 12 = 21.

So in this case, the relation between the two would be x + 2y = 21.
Again, the usual equation has y in terms of x,
so subtract x from both sides, giving 2y = 21 - x.
Now divide the entire equaion by 2, giving y = (21 - x)/2.

Lets put in 1, 5, and 9 for x and see what we get.

The first x, 1, gives (21 - 1)/2 = 20/2 = 10,
which is the first y value.

The second x, 5, gives (21 - 5)/2 = 16/2 = 8,
which is the second y value.

The third x, 9, gives (21 - 9)/2 = 12/2 = 6,
which is the third y value.

Wasn't that exciting?  If you want even more, go ahead and ask.
The next one that could be given to you might involve y = x*x.


---------- FOLLOW-UP ----------

QUESTION: Thank you so much, I noticed x was increasing by 4 when y was decresing by 4 but I did not know what to do with it.  To further understand I read your example. I don't understand "Note that here, every time x increases by 4, y decreases by 2.
This says multiply the y value by 2 and add to the x."
Can you explain this step furthur

Answer
I'm not sure if I answered this question, but will respond to it.

Since x goes from 1 to 5 and then 5 to 9, the increase is 5-1 = 9-5 = 4.
Since y goes form 10 to 8 and 8 to 6, that is 8-19 - 6-8 = -2.

To get the slope of x with respect to y, it is 4/(-2), which is -2.

To get an equation, use x - x1 = m(y - y1).
That is, x - 1 = -2(y - 10)
The equation would be found by multiply this out.
This gives x - 1 = -2y + 20, which is the same as x = -2y + 21.
Thus, where y = 10, x = -20 + 21 = 1; when y = 8, x = -16 + 21 = 5,
and when y = 6, x = -12 + 21 = 9.