Hi! I'm happy to help! (sorry for the wait, my computer crashed)
A linear function is where one variable is independent (x), and another is dependent. This means that we use x to solve for y, y depends on x to be found.
To solve this, we use an equation call slope intercept form, which states that y=mx+b.
There is only 1 solution line. (to solve for how to find y, you only have to have two verifications)
For our first table, we have x as 1, 2, 3,and 4. y is 3, 6, 12, and 24.
For our first set, with x is one and y is 3, there are infinite solutions, so we have to test and see if it matches the second set of x is 2 and y is 6.
How I like to check for what m will be, I check the first two y values. The first two are 3 and 6. They have an increase of 3, so m will be 3. (this only works when x increases by 1)
With m being 3, we can solve for b, to complete the equation:
3=(3(1))+b
3=3+b
With this, b must equal 0
3=3+0
Now, we just have to check if this equation matches all of the other x values:
6=(3(2))+0
6=6+0
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12=(3(3))+0
12=9+0
12≠9
This equation doesn't seem to work, so this is not a linear function.
For our next table, our first two y values are 2 and 5, which have a increase of 3, so our m will be 3 again.
Now, let's solve for b:
2=(3(1))+b
2=3+b
In this case, b will be -1 because 2=3-1
Now, let's check if this matches up with our other values:
5=((3)2)-1
5=6-1
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9=((3(3)-1
9=9-1
9≠8
This equation doesn't work, so it is not a linear function.
For our next table, we have our first two y values as -3, and -5. This has a decrease of 2, so our m will be -2.
Now, let's solve for b:
-3=((-2)1)+b
-3=-2+b
In this equation, b will be -1 because -3=-2-1.
Now, let's check our other values:
-5=(-2(2))-1
-5=-4-1
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-7=(-2(3))-1
-7=-6-1
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-9=(-2(4)-1
-9=-8-1
All of these values match, so this is a linear function.
If you wanted to (you don't have to, but if you needed verification) you could check the fourth problem.
Has a decrease of 2, so m would be -2. Plug it in:
-2=(-2(1))+b
-2=-2+b
b would be 0.
Check it on the second problem:
-4=(-2(2))+0
-4=-4+0
Check the third:
-2=(-2(3))+0
-2=-6+0
-2≠-6
The fourth table doesn't match either, so it is not a linear function.
You should choose option 3 because it is a linear function.
I hope this was helpful, keep learning! :D