Line of best fit y = 2x + 3, point (5, 10), find the residual

Step by step, with the reason for every step · Residuals · Statistics

Step by step

Question: find the residual for the line of best fit y = 2x + 3 and the point (5, 10).

The point is the real data.

y = 2x + 3; (5, 10)

A residual tells how far a point is from the line.

Residual = actual minus predicted.

Actual is the real y.

Predicted is the y the line gives.

residual = actual − predicted

We put x = 5 into the line.

It predicts y = 13.

predicted = 2(5) + 3 = 10 + 3 = 13

Careful!

Not predicted minus actual.

That flips the sign.

Actual comes first.

Careful: not 13 − 10 = 3 ✗

The actual y is 10.

10 minus 13 is −3.

residual = 10 − 13 = −3

So the residual is −3.

It is negative: the point is 3 below the line.

Let's check with a picture.

Each square up is 2.

The line of best fit.

The point.

The residual is the distance up or down from the line to the point.

And predicted plus residual gives the actual y: 10.

So the answer is right!

13 − 3 = 10

Let's look back.

First: predict.

Then: actual minus predicted.

Last: check.

Answer

residual = 10 − 13 = −3

Try this one yourself, then check:

Find the residual for the line of best fit y = 2x + 6; (5, 13)

Show the answer

residual = −3