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.

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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
The values of x were 1, 5, and 9.
The values of y were 10, 8, and 6.

Note that the numbers must be kept in that order,
so x = (1, 5, 9) and y = (10, 8, 6).

What I mean is that if numbers are looked at in order, the x's increa by 4 (5-1=4, 9-5=4) and the y's decrease by 2 (10-8=2, 8-6=2).

If we want the change for x and y to be the same,
add x and the double of y.  That is, x + 2y.

Note that for the first set (x,y) we have (1,10), which means that x is 1 and y is 10.  In this case, when we look at x + 2y, we have
1 + 2*10 = 1 + 20 = 21.

The second set of number, (5,8) means that x = 5 and y = 8.
Again, we look at x + 2y = 5 + 2*8 = 5 + 16 = 21.

And lastly, now the third set of numbers, (9,6).  Here, x=9 and y=6.
This time x + 2y is the same as 9 + 2(6) = 9 + 12 =21.

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So if we had the set X as {3, 6, 9, 12} and
the set Y as {31, 25, 19, 13},
then what should be noticed is that the difference in the X values is that they increase by 3 each time - 3+3=6, 6+3=9, 9+3=12.

The difference in the Y values is that they decrease by 6 each time:
31-25 = 6, 25-19=6, 19-13=6.

Since 6 is twice 3, this means that y is decreasing twice as fast as x is increasing.  To make x match up to y, take 2x + y.

The first element in each set, x=3 and y=31,
then go 2(3) + 31 = 6 + 31 = 37.

The 2nd number in each set, x=6 and y=25,
then go 2(6) + 25 = 12 + 25 = 37.

The 3rd number in eash set, x=9 and y=19,
then go 2(9) + 19 = 18 + 19 = 37.

The 4th number in each set, x=12 and y=13,
then go 2(12) + 13 = 24 + 13 = 37.

So the equation here is 2x + y = 37, or y = 37 - 2x.


So do you see what Y would be if x were 1?    Take y = 37 - 2x.
Well, y = 37 - 2(1) = 37 - 2 = 35.

How about x = 13?  Take y = 37 - 2x.
Well, y = 37 - 2(13) = 37 - 26 = 11.

Can you do x = 5 in this example?  How about x = 18?

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Now what if y went 1, 2, 3, 4 and x went 1, 4, 9, 16.

In this case, y = x*x = x².

Do you understand this one?

What if x were 5?  What if x were 13?