1. Straight Lines

2. 1. Horizontal Line y = c ; c is a constant Example: y – 5 = 0 y = 5 ( Dom & Range of this function?)

3. 2. Vertical Line x = c ; c is a constant Example: x – 5 = 0 x = 5 Not a function (Why?)

4. 3. Line through the Origin y = cx ; c is a constant Example: y – 5x = 0 y = 5x (Domain & Range of this function?)

5. 4. Line intersecting both axis at distinct points y = ax+b ; c is a constant Examples: 2y - 4x= 12 y = 6 + 2x ( Domain & Range of this function?)

6. Slope from two Points of the Line
The slope m of a non-vertical straight line through the points (x1 , y1) and (x2 , y2) is:
m = (y2 - y2) / (x1 , x2)
Find, if exists, the slope of the line through:
1. (5,6) and (5,7)
2. (5,6) and (2,6)
3. (4,3) and (8,5)
3. (5,10) and (6,12)

7. Solution 1. The slope does not exist. Why? 2. The slope is zero. Why?
Is the slope of every horizontal line equal to zero?
3. The slope =(5-3)/(8-4)= 2/4=1/2
4. The slope = (12– 10)/(6-5) = 2/1 = 2

8. The slope of a non-vertical straight line from the Equation
The slope m of a non-vertical straight having the equation ax + by + c = 0 ( What can you say about b?) is; m = - a / b
Find, if exists, the slope of the given line :
1. x = 5
2. y = 6
3. 2y - 1 = x
4. y = 2x.

9. Solution
1. The slope does not exist. Why?
2. The slope is zero. Why?
Is the slope of every horizontal line equal to zero?
3. Rewrite the equation in the general form:
First rewrite the equation in the general form: 2y - 1 = x → x – 2y + 1 = 0
The slope = - 1 / (-2) = 1 / 2
4. Rewrite the equation in the general form:
y = 2x → 2x – y = 0
The slope = - 2 / (-1) = 2

10. Finding the Equation from the Slope and a point of the Line
1. The equation of a non-vertical straight line having the slope m and through the point (x0, y0) is:
y - y0 = m ( x - x0)
Find, the equation of the straight line through the point (2 , 4) and:
1. Having no slope
2. Having the slope 0
3. Having the slope 3

11. Solution 1. x = 2 (Why?) 2. y = 4 (Why?) 3. y – 4 = 3 (x – 2)

12. Finding the Equation from Two Points of the Line
1. Find, the equation of the straight line through the point (2 , 4) and ( 2 , 5)
2. Find, the equation of the straight line through the point (2 , 4) and ( 5 , 4)
3. Having the slope (2 , 4) and ( 5 , 10)

13. Solution
1. This is a vertical line. What’s the equation?
2. This is a horizontal line. What’s the equation?
3. We find the slope from the two points, and then use that together with one point to find the equation.

14. Finding the Equation from the Slope & the y-intercept
1. Find, the equation of the straight line with slope 5 and the y-intercept 3
2. Find, the equation of the straight line with slope 5 and y-intercept - 3
3. Find, the equation of the horizontal straight line y-intercept 3
4. Find, the equation of the straight line with slope 5 and the y-intercept 0

15. Solution
1. This is line with slope 5 and through the point (0,3). What’s the equation of this line?
2. This is line with slope 5 and through the point (0,-3). What’s the equation of this line?
3. This is line through the point (0,3). What’s the equation of a horizontal line through the point (0,3)?
3. This is line with slope 5 and through the point (0,0). What’s the equation of this line?

16. Parallel & Perpendicular Lines
1. Two non-vertical straight lines are parallel iff they have the same slope.
2. Two non-vertical non-horizontal straight lines are perpendicular iff they the slope of each one of them is equal negative the reciprocal of the slope of the other.
3. A vertical line is parallel only to a vertical line.
4. A vertical line is perpendicular only to a horizontal line or visa versa

17. Parallel & Perpendicular Lines
That’s:
I. if a line L1 and L2 have the slopes m1 and m2 respectively, then:
1. L1 // L2 iff m1 = m2
2. L1 ┴ L2 iff m2 = - 1 / m1
II.
a. If a line L parallel to vertical line, then L is vertical
b. If a line L perpendicular to vertical line, then L is horizontal
c. If a line L perpendicular to horizontal line, then L is vertical

18. Examples (1) a. The following pair of lines parallel (Why?)
1. The line x - 2y + 8 = 0 and the line 10y = 5x-12
2. The line 3 - 2x = 0 and the line x = √2
3. The line 7y + 4x = 0 and the line 9 – 14y =8x
3. The line 7y – 4 = 0 and the line y = π
b. The following pair of lines perpendicular (Why?)
1. The line x - 2y + 8 = 0 and the line 5y = -10x - 12
2. The line 3 - 2x = 0 and the line y = √2
3. The line 2x +3y + 8 = 0 and the line 4y = 6x + 7

19. Examples (2)

20. Solution