About this video
Sal checks whether a table with points (-7, 4), (-3, 3), (1, 2) and (7, 1) is linear by comparing change in y over change in x. The first two pairs give -1/4, but the last pair gives -1/6, so the ratio isn't constant and the table is not a linear equation.
You'll be able to
- find the change in x and change in y between two points using delta notation.
- calculate change in y over change in x for each pair of points.
- decide if a table is linear even when the x-steps are not equal.
Key words
- delta (Δ)
- A Greek letter used as shorthand for 'change in'.
- rate of change
- Change in y divided by change in x between two points.
- linear equation
- An equation whose change in y over change in x is the same constant for any two points.
Transcript
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.
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