Consider the points R and S in the figure. How many different lines pass through both R and S? Explain.

Here we have two points R and S.

How many lines pass through both R and S?

Well, we can have only one line that passes through both R and S.

Because we are working with lines, we can only see a plane X-Y here.

Then the points R and S can be written as:

R = (x1, y1) and S = (x2, y2)

Now, a line or a  linear relationship can be written as:

y = a*x + b

where a is the slope and b is the y-axis intercept.

For a line that passes through the points (x1, y1) and (x2, y2), the slope can be written as:

a = (y2 - y1)/(x2 - x1).

Now, let's suppose that we have two different lines that pass through points R and S.

From above, both lines must have the same slope, because they pass through the same points.

y1 = a*x + b

y2 = a*x + c

So the only thing that our lines can have different is the y-intercept, but the y-intercept acts as a vertical shift.

This means, if we have different values in the y-intercept, we will have two parallel lines.

And two parallel lines can never pass through the same points, which means that we can not have different values for the y-intercept.

Then the two lines must be identical.

Then there is only one line that passes through points S and R at the same time,

good luck on your studies! : )

david edward.