A line passes through the points A(-3,-2) and B(2,1). Does it also pass through the point C(5,3)?

Question

Justify how you know. 

Answer 1

Given the two points A=(-3,-2) and B =(2,1) we can find

the equation of the line through points A, B

 

The slope = rise/run = [1-(-2)]/[2-(-3)] = 3/5

So the equation of the line in slope intercept form of y=mx+b

looks like  y = (3/5)x + b

 

We can use one of the given points to solve for b

From point B, x =2 and y = 1

1 = (3/5)(2) + b

1 = 6/5 + b

b = -1/5

 

Note:  I also worked the problem to find b using

point A and got the same answer, as I should...

 

The equation of the line through AB is:

y = (3/5)x -1/5

 

We need to know if the point C=(5,3) is on the line.

If so then we should have a valid equation when we

substitute in x = 5  and  y = 3 into the equation.

 

3 = (3/5)(5)-1/5

3 = 3-1/5

3 = 2 4/5

 

This is not valid, therefore point C=(5,3) is not on the line.

Answer 2

The equation of the line passing through points A and B is:

 

Slope = (-2-1)/(-3-2) = 3/5

 

The equation of the line calculated from point A is :

 

-2 = 3/5 (-3) + b

 

intercept b = -2 + 9/5 = -1/5

 

equation is y = 3/5 X - 1/5

 

Plugging point C into the line gives:

 

3 = (3/5) * 5 -1/5

   = 3 -1/5 = 14/5

 

The sides of the equation are not equal, then the line passing through A and B does not pass through C.