Equations of Lines

I.  Homework

II.  Parallel and Perpendicular Lines

Definition:  

A)  Two lines are parallel if they have the same slope. 

B)  Two lines are perpendicular if their slopes are negative reciprocals of each other.  That is

if m₁ is the slope of the first line and m₂ is the slope of the second line, then

m₁ = -(1/m₂)  

Example:  Determine if the lines that pass through the given points are parallel, and which are perpendicular.

Line 1:  P = (2,3)  Q = (1,5)

Line 2:  P = (4,6) Q = (2,5)

Line 3:  P = (-3,2) Q = (1,4)

V.  Equations of Lines

If we are given two points, how can we find the equation of the line passing through the two points?  If we know the slope and a point (x₁, y₁) we have that

m = rise/run = (y - y₁)/(x - x₁)    so that

Point Slope Formula for the Equation of a Line:  y - y₁ = m(x - x₁) 

Example:  Find the equation of line through the point (1,2) with slope 4.

Solution:  We use the formula:  y - 2 = 4(x - 1) = 4x - 4

Hence y = 4x - 4 + 2 = 4x - 2.

Step by Step Procedure:

Step 1:  Use the slope formula to find the slope m.

Step 2:  Plug the first point and the slope into the point slope formula.

Step 3:  Simplify to get into the form y = mx + b.

Example:  Find the equation of the line passing through the points

(1,4) and (2,9)

1)  We find m = (9- 4)/(2 - 1) = 5

2)  We plug into the formula:

y - 4 = 5(x - 1)

3)  We have y - 4 = 5x - 5

y = 5x - 1.

We see that if the equation is in the form y = mx + b them m is the slope and b is the y intercept.

In the above example, the slope is 5 and the y intercept is -1.

Exercises:  

A)  Find the euquation of the line through the point (1,2) that is perpendicular to the

line y = 3x - 4

B)  Find the equation of the line through the point (3,4) that is parallel to the line

3x - 2y = 5

Application:  At your hotel if you charge $100 per night you can fill 80 of your rooms while if you charge $90 per night you can fill 85 of your rooms.  Assume that the vacancy and the price are linearly related.  

A)  Come up with an equation that predicts how many rooms y you will fill for a given price x.

B)  How many rooms will you fill if you charge $120 per night.

Solution:

A)  We have the two points (100,80) and (90, 85)  

We compute the slope:  m = (85 - 80)/(90 - 100) = 5/-10 = -1/2

Then we use the point slope formula:

y - 80 = -1/2 (x - 100)

y - 80 = -1/2 x +50

y = -1/2 x + 130

B)  We plug in 120:

y = -1/2 (120) + 130 = 70

Therefore, at $120 we can expect to fill 70 of our rooms.