Thread: slopes

Each pair of points is on a line. What is the slop of each line?
A(7,5), B(2,4)

A(5,2), B(2,-1)

A(-3,3) B(5,3)

Write the equation for the line passing through each pair of points

(3,8), (2,6)

(-2,-4), (5,-1)


Can someone show me how to do all of these step by step?

To calculate the slope, we use the formula (y2-y1)/(x2-x1)

For the first set:

A = (7,5) = (x1,y1), B = (2,4) = (x2,y2)

=> slope = (4 - 5)/(2 - 7) = -1/-5 = 1/5


For the second set:

A = (5,2) = (x1,y1), B = (2,-1) = (x2,y2)

=> slope = (-1 - 2)/(2 - 5) = -3/-3 = 1


For the third set:

A = (-3,3) = (x1,y1), B = (5,3) = (x2,y2)

Here we notice the y value is the same in both coordinates, it means therefore that a horizontal line connects these two points, so the slope is 0. Nevertheless, let's go through the routine.

=> slope = (3 - 3)/(5 + 3) = 0/8 = 0........as expected



Now for the equation of a line section.

Recall that the equation of a line is in the form y = mx + b, where m is the slope and b is the y-intercept. So first we need to find the slope of the line connecting the pair of points, then we can plug it into the above formula to find b and rewrite the formula in the form above, OR we can plug the values we know into the formula y - y1 = m(x - x1), where x1 and y1 come from any one of the points given, then you just solve for y, and your equation will be in the above form--I usually use the latter approach, and that's the one I will be using. Here goes.

For (3,8) , (2,6)

slope = m = (6 - 8)/(2 - 3) = -2/-1 = 2

Using (x1,y1) = (3,8)

y - y1 = m(x - x1)

=> y - 8 = 2(x - 3)
=> y = 2x -6 + 8
so y = 2x + 2


For (-2,-4) , (5,-1)

slope = m = (-1 + 4)/(5 + 2) = 3/5

Using (x1,y1) = (5,-1)

y - y1 = m(x - x1)

=> y + 1 = (3/5)(x - 5)
=> y = (3/5)x -3 - 1
so y = (3/5)x - 4