Equations of Lines Parallel and Perpendicular Lines
Remark: If m_{1} is the slope of the first line and m_{2} is the slope of the second line, then
1 We can also say that two lines are perpendicular if m_{1}m_{2} = 1 This definition does not apply to vertical and horizontal lines. Any two distinct vertical lines are parallel and any two distinct horizontal lines are also parallel. Any vertical line is perpendicular to any horizontal line.
Example Line 1: P = (2,3) Q = (1,5) Line 2: P = (4,6) Q = (2,5)
Line 3: P = (3,2)
Q = (1,4) Solution We compute the slopes of the three lines
5  3
5 
6
1
4 
2
1 Since m_{2} = m_{3} Line 2 and Line 3 are parallel. Since m_{1}m_{3} = 1 Line 1 and Line 3 are perpendicular. Also Line 1 and Line 2 are perpendicular. 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 (x_{1}, y_{1}) we have that
rise
y  y_{1} Now multiplying by x  x_{1} on both sides of the equation results in the following formula.
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).
We see that if the equation is in the form y = mx + b then 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
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.
Solution:
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