1. 4.1 Introduction to Linear Equations in Two Variables
A linear equation in two variables can be put in the form (called standard form): where A, B, and C are real numbers and A and B are not zero

2. 4.1 Introduction to Linear Equations in Two Variables
Table of values (try to pick values such that the calculation of the other variable is easy): x y 6 2 3 4

3. 4.1 Introduction to Linear Equations in Two Variables
Points: (2, 3) 2 is the x-coordinate, 3 is the y-coordinate
Quadrants:
II x<0 and y>0
I x>0 and y>0
III x<0 and y<0
IV x>0 and y<0

4. 4.2 Graphing by Plotting and Finding Intercepts
The graph of any linear equation in two variables is a straight line. Note: Two points determine a line.
Graphing a linear equation:
Plot 3 or more points (the third point is used as a check of your calculation)
Connect the points with a straight line.

5. 4.2 Graphing by Plotting and Finding Interceptsx y 6 2 3 4
Graph:

6. 4.2 Graphing by Plotting and Finding Intercepts
Finding the x-intercept (where the line crosses the x-axis): let y = 0 and solve for x
Finding the y-intercept (where the line crosses the y-axis): let x = 0 and solve for y Note: the intercepts may be used to graph the line.

7. 4.2 Graphing by Plotting and Finding Intercepts
If y = k, then the graph is a horizontal line:
If x = k, then the graph is a vertical line:

8. 4.2 Graphing by Plotting and Finding Interceptsx y -3 2 4
Example: Graph the equation.

9. 4.3 The Slope of a Line
The slope of a line through points (x1,y1) and (x2,y2) is given by the formula:

10. 4.3 The Slope of a Line A positive slope rises from left to right.
A negative slope falls from left to right.

11. 4.3 The Slope of a Line If the line is horizontal, m = 0.
If the line is vertical, m = undefined.

12. 4.3 The Slope of a Line Finding the slope of a line from its equation
Solve the equation for y.
The slope is given by the coefficient of x
Example: Find the slope of the equation.

13. 4.4 The Slope-Intercept Form of a Line
Standard form:
Slope-intercept form: (where m = slope and b = y-intercept)

14. 4.4 The Slope-Intercept Form of a Line
Example: Put the equation 2x + 3y = 6 in slope-intercept form, determine the slope and intercept, then graph. Since b = 2, (0,2) is a point on the line. Since , go down 2 and across 3 to point (3,0) a second point on the line, then connect the two points to draw the line.

15. 4.4 The Slope-Intercept Form of a Linex y 2 3
Example: Graph the equation.

16. 4.5 Writing an Equation of a Line
Standard form: Definition is now changed as follows: A, B, and C must be integers with A > 0
Slope-intercept form: (where m = slope and b = y-intercept)
Point-slope form: for a line with slope m going through point (x1, y1).

17. 4.5 Writing an Equation of a Line
Example: Find the equation of a line going through the point (2,5) with slope = 3. Express your answer in slope-intercept form. Start with the point-slope equation: Solve for y to get in slope intercept form:

18. 4.5 Writing an Equation of a Line
Example: Find the equation of a line going through the points (-3,5) and (0,3). Express your answer in standard form. Solve for the slope: Use slope intercept form & multiply by the LCD:

19. 4.6 Parallel and Perpendicular Lines
Parallel lines (lines that do not intersect) have the same slope.
Perpendicular lines (lines that intersect to form a 90 angle) have slopes that are negative reciprocals of each other.
Horizontal lines and vertical lines are perpendicular to each other

20. 4.6 Parallel and Perpendicular Lines
Example: Determine if the lines are parallel, perpendicular or neither: get the slope of each line the slopes are negative reciprocals of each other so the lines are perpendicular

21. 4.6 Parallel and Perpendicular Lines
Example: Find the equation in slope-intercept form of a line passing through the point (-4,5) and perpendicular to the line 2x + 3y = 6 (solve for y to get slope of line) (take the negative reciprocal to get the  slope)

22. 4.6 Parallel and Perpendicular Lines
Example (continued): Use the point-slope form with this slope and the point (-4,5) In slope intercept form: