1. LINEAR EQUATIONS MSJC ~ Menifee Valley Campus
Math Center Workshop Series
Janice Levasseur

2. Equations of the Line
Write the equation of a line given the slope and the y-intercept
Write the equation of a line given the slope and a point
Write the equation of a line given two points

3. Equations of the Line
Write the equation of a line given the slope and the y-intercept: m and (0, b)
Write the equation of a line given the slope and a point
Write the equation of a line given two points

4. Ex: Find an equation of the line with slope = 6 and y-int = (0, -3/2)
Recall: slope-intercept form of a linear equation
y = mx + b, where m and b are constants
Given the y-int = (0, -3/2)  b = - 3/2
Given the slope = 6  m = 6
Putting everything together we get the equation of the line in slope-int form: y m 6 x +
- 3/2 b =
y = 6x – 3/2

5. Ex: Find an equation of the line with slope = 1. 23 and y-int = (0, 0
Recall: slope-intercept form of a linear equation
y = mx + b, where m and b are constants
Given the y-int = (0, 0.63)  b = 0.63
Given the slope = 1.23  m = 1.23
Putting everything together we get the equation of the line in slope-int form: y m + b =
1.23 x
0.63
y = 1.23x

6. Equations of the Line
Write the equation of a line given the slope and the y-intercept
Write the equation of a line given the slope and a point: m and (x1, y1)
Write the equation of a line given two points

7. Ex: Find an equation of the line with slope = -3 that contains the point (4, 2)
Start with the slope-intercept form of a linear equation y = mx + b
Slope = - 3 
y = - 3x + b
What is b, though?
What is b, though?
Use the given point (4, 2)  x = 4 and y = 2
y = - 3x + b 
2 = - 3(4) + b
put it together
2 = b
we have m and b 
14 = b
y = - 3 x + 14

8. Ex: Find an equation of the line with slope = -0
Ex: Find an equation of the line with slope = that contains the point (2, -6)
Start with the slope-intercept form of a linear equation y = mx + b
Slope = 
y = x + b
What is b, though?
Use the given point (2, -6)  x = 2 and y = -6
y = x + b 
-6 = -0.25(2) + b
put it together
-6 = b
we have m and b 
-5.5 = b
y = -0.25x – 5.5

9. Equations of the Line
Write the equation of a line given the slope and the y-intercept
Write the equation of a line given the slope and a point
Write the equation of a line given two points: (x1, y1) and (x2, y2)

10. Ex: Find an equation of the line containing the points (-2, 1) and (3, 5)
First, find the slope of the line containing the points:
Slope = m = rise = y1 - y2 = 1 – (5)
run x1 - x – 3
Now we have m = 4/5 and two points. Pick one point and proceed like in the last section.

11. We have m = 4/5, the point (-2, 1), and y = mx + b
Slope = 4/5 
y = 4/5x + b
What is b, though?
Use the given point (-2, 1)  x = -2 and y = 1
y = 4/5x + b 
1 = 4/5(-2) + b
1 = (-8/5) + b
13/5 = b
put it together 
we have m and b 
y = 4/5x + 13/5

12. Ex: Find an equation of the line containing the points (-4, 5) and (-2, -3)
First, find the slope of the line containing the points:
Slope = m = rise = y1 - y2 = 5 – (-3)
run x1 - x – (-2)
= -4
Now we have m = -4 and two points. Pick one point and proceed like in the last section.

13. We have m = -4, the point (-4, 5), and y = mx + b
Slope = -4 
y = -4x + b
What is b, though?
Use the given point (-4, 5)  x = -4 and y = 5
y = -4x + b 
5 = -4(-4) + b
5 = 16 + b
-11 = b
put it together 
we have m and b 
y = -4x – 11

14. Ex: Find an equation of the line containing the points (0, 0) and (1, -5)
First, find the slope of the line containing the points:
Slope = m = rise = y1 - y2 = 0 – (-5)
run x1 - x – (1)
= -5
Now we have m = -5 and two points. Pick one point and proceed like in the last section.

15. We have m = -5, the point (0, 0), and y = mx + b
Slope = -5 
y = -5x + b
What is b, though?
Use the given point (0, 0)  x = 0 and y = 0
y = -5x + b 
0 = -5(0) + b
0 = 0 + b
0 = b
put it together 
we have m and b 
y = -5x + 0
y = -5x

16. Equations of the Line
Write the equation of a line given the slope and the y-intercept
Write the equation of a line given the slope and a point
Write the equation of a line given two points

17. Parallel & Perpendicular Lines
When we graph a pair of linear equations, there are three possibilities:
the graphs intersect at exactly one point
the graphs do not intersect
the graphs intersect at infinitely many points
We will consider a special case of situation 1 and also situation 2.

18. Perpendicular Lines (Situation 1)
Perpendicular lines intersect at a right angle
Notation:
L1: y = m1x + b1
L2: y = m2x + b2
L1 ^ L2

19. Nonvertical perpendicular lines have slopes that are the negative reciprocals of each other:
m1m2 = -1 ~ or ~ m1 = - 1/m2 ~ or ~ m2 = - 1/m1
If l1 is vertical (l1: x = a) and is perpendicular to l2, then l2 is horizontal (l2: y = b) ~ and ~ vice versa

20. Ex: Determine whether or not the graphs of the equations of the lines are perpendicular: l1: x + y = 8 and l2: x – y = - 1
First, determine the slopes of each line by rewriting the equations in slope-intercept form:
l1: y = - x + 8 and l2: y = x + 1 1 1
m1 = and m2 = -1 1
Since m1m2 = (-1)(1) = -1, the lines are perpendicular.

21. Ex: Determine whether or not the graphs of the equations of the lines are perpendicular: l1: -2x + 3y = -21 and l2: 2y – 3x = 16
First, determine the slopes of each line by rewriting the equations in slope-intercept form:
l1: y = (2/3)x - 7 and l2: y = (3/2)x + 8
m1 = and m2 =
2/3
3/2
Since m1m2 = (2/3)(3/2) = 1 = -1
Therefore, the lines are not perpendicular!

22. Parallel Lines (Situation 2)
Parallel lines do not intersect
Notation:
L1: y = m1x + b1
L2: y = m2x + b2
L1 || L2

23. Nonvertical parallel lines have the same slopes but different y-intercepts:
m1 = m2 ~ and ~ b1 = b2
Horizontal Parallel Lines have equations
y = p and y = q where p and q differ.
Vertical Parallel Lines have equations
x = p and x = q where p and q differ.

24. Ex: Determine whether or not the graphs of the equations of the lines are parallel: l1: 3x - y = -5 and l2: y – 3x = - 2
First, determine the slopes and intercepts of each line by rewriting the equations in slope-intercept form:
l1: y = 3x + 5 and l2: y = 3x - 2
m1 = and m2 = 3 3
b1 = and b2 = 5 -2
Since m1 = m2 and b1 = b2
the lines are parallel.

25. Ex: Determine whether or not the graphs of the equations of the lines are parallel: l1: 4x + y = 3 and l2: x + 4y = - 4
First, determine the slopes and intercepts of each line by rewriting the equations in slope-intercept form:
l1: y = -4x + 3 and l2: y = (-¼)x - 1
m1 = and m2 = -4
- ¼
Since m1 = m2
the lines are not parallel.