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CCSS Math: 8.F.A.3

Does the following table
represent a linear equation? So let's see what's
going on here. When x is negative 7, y is 4. Then when x is
negative 3, y is 3. So let's see what happened
to what our change in x was. So our change in
x-- and I could even write it over here,
our change in x. So going from negative
7 to negative 3, we had an increase in 4 in x. And what was our change in y? And this triangle, that's
just the Greek letter delta. It's shorthand for "change in." Well, our change in y
when x increased by 4, our y-value went from 4 to 3. So our change in
y is negative 1. Now, in order for this
to be a linear equation, the ratio between our change
in y and our change in x has to be constant. So our change in
y over change in x for any two points
in this equation or any two points in the table
has to be the same constant. When x changed by 4, y
changed by negative 1. Or when y changed by
negative 1, x changed by 4. So we have to have a
constant change in y with respect to x
of negative 1/4. Let's see if this is true. So the next two points, when
I go from negative 3 to 1, once again I'm
increasing x by 4. And once again, I'm
decreasing y by negative 1. So we have that same ratio. Now, let's look at
this last point. When we go from 1 to
7 in the x-direction, we are increasing by 6. And when we go from 2 to 1,
we are still decreasing by 1. So now this ratio, going
from this third point to this fourth point,
is negative 1/6. So it is not. So just for this last
point right over here, for this last point, our change
in y over change in x, or I should say, really, between
these last two points right over here, our change
in y over change in x-- let me clear this up. Let me make it clear. So just between these
last-- in magenta. Just between these last
two points over here, our change in y is negative
1, and our change in x is 6. So we have a different rate of
change of y with respect to x. Because we had a different
rate of change of y with respect to x, or ratio
between our change in y and change in x, this is
not a linear equation. No, not a linear equation.