Section 3.2 Special Forms of Linear Equations in Two Variables

3.2 Lecture Guide: Special Forms of Linear Equations in Two Variables Objective 1: Use the slope-intercept form to write and graph linear equations.

Algebraically Algebraic Example Verbal Example Graphically y = mx + b is the equation of a line with _______ m and y-intercept ________. This line has a slope of and a y-intercept of (0, 3). Slope-Intercept Form

1. The linehas a slope of ____________ and a 2. Graph the line using the slope and the y-intercept. y-intercept of ____________.

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. Slopey-interceptEquation

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.

Complete the missing information. On all graphs, clearly label at least two points. 4. Equation: Through the point: Slope: Graph:

5. Equation: Through the point: Slope: Complete the missing information. On all graphs, clearly label at least two points. Graph:

Complete the missing information. 6. Graph: Through the point: Slope: Equation:

7. Graph: Through the point: Slope: Equation: Complete the missing information.

8. Use the given table of values for a linear function to determine the following. (a) The value of. (b) The value of. (c) The slope of this line. (d) The y-intercept of this line. (e) The equation of this line in slope- intercept form.

Objective 2: Use the point-slope form to write and graph linear equations. Algebraically Algebraic Example Graphical Example Verbal Example is the equation of a line through __________ with ________ m This line passes through The point (1, 1) with a slope of. Point-Slope Form:

9. 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. SlopePointPoint-Slope Equation

10. Through the point: Slope: Graph: Complete the missing information. On all graphs, clearly label at least two points.

11. Through the point: Slope: Graph: Complete the missing information. On all graphs, clearly label at least two points.

Write each equation in slope-intercept form. 12.

Write each equation in slope-intercept form. 13.

Write the equation of the line passing through the given point with specified slope. Write the answer in slope-intercept form. 14. (2, – 3), m = – 3

Write the equation of the line passing through the given point with specified slope. Write the answer in slope-intercept form. 15. ( – 3, 2),

Write the equation of the line passing through the given point with specified slope. Write the answer in slope-intercept form. 16. (– 4, 2),

Write the equation of the line passing through the given point with specified slope. Write the answer in slope-intercept form. 17. (4, – 3),

18. (– 2, 4) and (– 1, – 3) Write in slope-intercept form the equation of the line passing through the given points.

19. (– 4, – 4) and (1, 2) Write in slope-intercept form the equation of the line passing through the given points.

Use the given graph to complete the missing information. 20. Graph: Through the point: Slope: Slope-intercept equation:

21. Graph: Through the point: Slope: Use the given graph to complete the missing information. Slope-intercept equation:

Algebraically Numerical Example Graphical Example Verbally y = b is the equation of a ________line with y-intercept (o, b). Example: y = 3 This horizontal line has a y- intercept of (0, 3) and a slope of 0. Objective 3: Use the special forms of equations for horizontal and vertical lines. Horizontal Lines

Algebraically Numerical Example Graphical Example Verbally x = a is the equation of a ________line with x-intercept (a, 0). Example: x = – 2 This vertical line has an x- intercept of (– 2,0) and its slope is undefined. Vertical Lines

22. 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 _____. 23. 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: 24. x-intercept: ______ y-intercept: ______ Slope: _______ Graph:

Equation: 25. x-intercept: ______ y-intercept: ______ Slope: _______ Graph:

Complete the missing information. On all graphs, clearly label at least two points. 26. Equation: Through the point: (– 3, 4) Slope: m = Equation: Through the point: (4, – 3) Slope: m is undefined Graph:

Complete the missing information. 28. Graph: Through the y-intercept: Slope: Equation: 29. Graph: Through the x-intercept: Slope: Equation:

Parallel lines have the ____________ slope and _____________________ lines have slopes that are opposite reciprocals. Parallel and Perpendicular Lines

Determine if the first line is parallel to, perpendicular to, or neither parallel nor perpendicular to the second line. 30.

Determine if the first line is parallel to, perpendicular to, or neither parallel nor perpendicular to the second line. 31.

Determine if the first line is parallel to, perpendicular to, or neither parallel nor perpendicular to the second line. 32.

33. Write in slope-intercept form the equation of a line passing through (– 2, – 3) and parallel to y = 5x – 2.

34. Write in slope-intercept form the equation of a line passing through (– 2, – 3) and perpendicular to y = 5x – 2.

General Form The general form, Ax + By = C, of an equation is useful for writing linear equations without fractions. It is customary to write equations in general form with a positive coefficient on x.

35. Write each equation in general form.

36. Write each equation in general form.

37. The given table displays the dollar cost of a collect phone call based on the length of the call in minutes. (a) Determine the linear equation for the line that contains these data points. (b) Determine the meaning of m and b in this application.

38. The given graph displays the dollar cost of having a clothes dryer repaired by a service shop. (a) Determine the linear equation for this line. (b) Determine the meaning of m and b in this application. Hours Cost ($)