Section 3.2

Special Forms of Linear

Equations in Two Variables

3.2 Lecture Guide: Special Forms of Linear Equations in Two Variables

Objective: Use the slope-intercept and point-slope forms of a linear equation.

Slope-Intercept Form

Algebraically y mx b is the equation of a linewith slope m and y-intercept 0,b .

Algebraic Example1

32

y x

Verbal Example

This line has slope 12

and a y-intercept of 0,3 .

Graphical Example

13

2y x

0,3

2,4

12

-5

5

-5 5

y

x

1. The line 2

45

y x has a

slope of ____________ and a y-intercept of ____________.

2. Graph the line 2

45

y x

using the slope and the y-intercept.

3. Complete the following table. This example stresses the fact that if we know the slope and the y-intercept, then we can immediately write the equation. Also, if we have the equation in slope-intercept form, we can immediately sketch the graph because we can quickly determine the slope and the y-intercept.

Slope y-intercept Equation

2

3 0,1

3 0, 4

57

3y x

15

3f x x

Point-Slope Form

1 1y y m x x Algebraically

is the equation of a line through 1 1,x y with slope m.

Algebraic Example

11 13

y x

Graphical Example

x

y

Verbal ExampleThis line passes through the point

1,1 with slope 13

.

1,1 4,2

13

4. Complete the following table. This example stresses the fact that the point-slope form of a line is useful when the slope and a point other than the y-intercept is given.

Slope Point Point-Slope Equation

2,14

3 2, 3

1

2 3,2

3, 4 2

5

2 3y x

34 1

2y x

General Form:

The general form, Ax By C , of an equation is useful forwriting linear equations without fractions.

Write each equation in general form.

5. 13 5

x y

Write each equation in general form.

6.1 4

3 3y x

Write each equation in slope-intercept form:

7. 4 3 18x y

y mx b

Write each equation in slope-intercept form:

8. 8 10 20x y

y mx b

Parallel and Perpendicular Lines:Recall that parallel lines have the same slope and that perpendicular lines have slopes that are opposite reciprocals.

Determine if the lines are parallel, perpendicular, or neither parallel nor perpendicular.

9. 32

4f x x and 3

14

f x x

Determine if the lines are parallel, perpendicular, or neither parallel nor perpendicular.

10. and 2 3y x 6 12 24x y

Determine if the lines are parallel, perpendicular, or neither parallel nor perpendicular.

11. and 3 2 8x y 6 4 12x y

Objective: Use the special forms of equations for horizontal and vertical lines.

Horizontal and Vertical Lines

Algebraically y b is the equation

0,b .

Example: 3y

Numerical Example

2 3

1 3

0 3

1 3

2 3

x y

Graphical Example

-4

4

-4 4

x

yof a horizontal line with y-intercept

Verbally This horizontal line has a y-intercept of 0,3and a slope of 0.

3y

-4

4

-4 4

x

y

Horizontal and Vertical Lines

Algebraically x a is the equation

,0a .

Example: 2x

Numerical Example

Graphical Example

of a vertical line with x-intercept

Verbally This vertical line has an x-intercept of 2,0and its slope is undefined.

2x

2 2

2 1

2 0

2 1

2 2

x y

12. All points on a horizontal line have the same _____-coordinate. This is the reason that the equation of a horizontal line is of the form _________________________. The slope of a horizontal line is _____.

13. All points on a vertical line have the same _____-coordinate. This is the reason that the equation of a vertical line is of the form _________________________. The slope of a vertical line is _________________________.

Graph each equation by completing a table of values and then give any intercepts. Can you check both of these on a graphing calculator? If not, why not?

Equation: 3x 14.

x y

x-intercept: ______

y-intercept: ______

Slope: _______ -5

5

-5 5

y

x

Graph:

Equation: 15.

-5

5

-5 5

y

x

x y

x-intercept: ______

y-intercept: ______

Slope: _______

2y

Graph:

Objective: Graph a line given one point and its slope.To graph a line or to give the equation of the line, it is sufficient to know any point on the line and the slope of the line. This is equivalent to knowing any two points on the line because we can calculate the slope given any two points on the line.

-5

5

-5 5

y

x

Complete the missing information. On all graphs, clearly label at least two points.

16. Equation:

Through:

Slope:

32

2y x

Graph:

-5

5

-5 5

y

x

Complete the missing information. On all graphs, clearly label at least two points.

17. Equation:

Through:

Slope:

2 3 6y x

Graph:

-5

5

-5 5

y

x

Complete the following table. On all graphs, clearly label at least two points.

18. Equation:

Through:

Slope:0m

3,4

Graph:

-5

5

-5 5

y

x

Complete the following table. On all graphs, clearly label at least two points.

19. Equation:

Through:

Slope:Undefined

4, 3

Graph:

-5

5

-5 5

y

x

Complete the following table. On all graphs, clearly label at least two points.

20. Equation:

Through:

Slope:

12 3

2y x

Graph:

-5

5

-5 5

y

x

Complete the missing information. On all graphs, clearly label at least two points.

21. Equation:

Through:

Slope:

3 2 1y x

Graph:

Write each equation in slope-intercept form.

22. 12 3

2y x

Write each equation in slope-intercept form.

23. 3 2 1y x

Use the point-slope form to write the equation passing through the given point with specified slope. Write the answer in slope-intercept form.

24. 4,2 , 34

m

Use the point-slope form to write the equation passing through the given point with specified slope. Write the answer in slope-intercept form.

25. , 4, 323

m

26. Write in slope-intercept form the equation of a line passing through 2,4 and 1, 3 .

27. Write in slope-intercept form the equation of a line passing through and . 4, 4 1,2

28. Write in slope-intercept form the equation of a line passing through 2, 3 and parallel to 5 2y x .

29. Write in slope-intercept form the equation of a line passing through 2, 3 and perpendicular to 5 2y x .

-5

5

-5 5

y

x

Complete the missing information.

30. Graph: Through:

Slope:

Equation:

-5

5

-5 5

y

x

Complete the missing information.

31. Graph: Through:

Slope:

Equation:

-5

5

-5 5

y

x

Complete the missing information.

32. Graph: Through:

Slope:

Equation:

-5

5

-5 5

y

x

Complete the missing information.

33. Graph: Through:

Slope:

Equation:

-5

5

-5 5

y

x

Complete the missing information.

34. Graph: Through:

Slope:

Point-Slope Equation:

Slope-Intercept Equation:

-5

5

-5 5

y

x

Complete the missing information.

35. Graph: Through:

Slope:

Point-Slope Equation:

Slope-Intercept Equation:

36. The given table displays the dollar cost of a collect phone call based on the length of the call in minutes.

Minutes Cost

3 $2.30

5 $3.00

10 $4.75

13 $5.80

x y (a) Determine the linear equation f x mx b for the line that contains

these data points.

(b) Determine the meaning of m and b in this application.

37. The given graph displays the dollar cost of having a clothes dryer repaired by a service shop.

020406080

100120140160180

0 1 2 3 4 5

x

y (a) Determine the linear equation f x mx b for this line.

(b) Determine the meaning of m and b in this application.

5, 180

1, 80

Hours

Co

st

($)