1. Cartesian Plane and Linear Equations in Two Variables
Math 021

2. The Cartesian Plane (coordinate grid) is a graph used to show a relationship between two variables.
The horizontal axis is called the x-axis.
The vertical axis is called the y-axis.
The point of intersection of the x-axis and y-axis is called the origin.
The axes divide the Cartesian Plane into four quadrants.
An ordered pair is a single point on the Cartesian Plane. Ordered pairs are of the form (x,y) where the first value is called the x-coordinate and the second value is called the y-coordinate.

3. Examples – Plot each if the following ordered pairs on the Cartesian Plane and name the quadrant it lies in:
a. A = (3, 4)
b. B = (-2, 1)
c. C = (7, -3)
d. D = (-4, -2)
e. E = (0, 5)
f. F = (-1, 0)

4. Linear Equations in Two Variables
A linear equation in two variables is an equation of the form Ax + By = C where A, B, and C are real numbers.
The form Ax + By = C is called the standard form of a linear equation in two variables.
An ordered pair is a solution to a linear equation in two variables if it satisfies the equation when the values of x and y are substituted.

5. Examples – Determine if the ordered pair is a solution to each linear equation:
a. 2x – 3y = 6; (6, 2)
b. y = 2x + 1; (-3, 5)
c. 2x = 2y – 4; (-2, -8)
d. 10 = 5x + 2y; (-4, 15)

6. Examples – Find the missing coordinate in each ordered par given the equation:
a. -7y = 14x; (2, __ )
b. y = -6x + 1; ( ____, -11)
c. 4x + 2y = 8; (1, __ )
d. x – 5y = -1; ( ____, -2)

7. Complete the table of values for each equation:
y = 2x – 10 x + 3y = 9 x y 4
-20 5 x y 6 4

8. Graphing Linear Equations in Two Variables
The graph of an equation in two variables is the set of all points that satisfies the equation.
A linear equation in two variables forms a straight line when graphed on the Cartesian Plane.
A table of values can be used to generate a set of coordinates that lie on the line.

9. Graph: 2x + y = 4

10. Graph: y= 3x-1

11. Graph: y= 2x

12. Graph: 15= -5y + 3x

13. Intercepts An intercept is a point on a graph which crosses an axis.
An x-intercept crosses the x-axis. The y-coordinate of any x-intercept is 0.
A y-intercept crosses the y-axis. The x-coordinate of any y-intercept is 0.

14. Graph by Finding Intercepts: 3x – 2y = 12

15. Graph by Finding Intercepts: y= -2x + y

16. Graph by Finding Intercepts: 4x + 3y = -12

17. Graph by Finding Intercepts: 3x – 5y = -15

18. Horizontal and Vertical Lines
A horizontal line is a line of the form y = c, where c is a real number.
A vertical line is a line of the form x = c, where c is a real number.

19. Graph: x = 4

20. Graph: y= -2

21. Graph: 3x = -15

22. Graph: y + 3 = 4

23. Slope of a Line
The slope of a line is the degree of slant or tilt a line has. The letter “m” is used to represent the slope of a line.
Slope can be defined in several ways:
Examples - Find the slope of each line:
a. Containing the points (3, -10) and (5, 6)
b. Containing the points (-4, 20) and (-8, 8)

24. Find the slopes of the lines below:

25. Slopes of Horizontal & Vertical Lines
The slope of any horizontal line is 0
The slope of any vertical line is undefined
Examples – Graph each of the following lines then find the slope
x= y -2 = 4

26. Slope-Intercept form of a Line
The slope-intercept form of a line is y = mx + b where m is the slope and the coordinate (0,b) is the y-intercept.
The advantage equation of a line written in this form is that the slope and y-intercept can be easily identified.

27. Examples – Find the slope and y-intercept of each equation:
a. y = 3x – 2
b. 4y = 5x + 8
c. 4x + 2y = 7
d. 5x – 7y = 11

28. Parallel and Perpendicular Slopes
Two lines that are parallel to one another have the following properties
They will never intersect
They have the same slopes
They have different y-intercepts
Parallel lines are denoted by the symbol //
Two lines that are perpendicular to one another have the following properties:
They intersect at a angle
The have opposite and reciprocal slopes
Perpendicular lines are denoted by the symbol ┴

29. Complete the following table:
Slope
// Slope
┴ slope a. b. 5 c.

30. Examples – Determine if each pair of lines is parallel, perpendicular, or neither:
a. 2y = 4x + 7 b. 5x – 10y = 6
y – 2x = y = 2x + 7
c. 3x + 4y = 3 d. Line 1 contains points (3,1) and (2,7)
4x + 5y = Line 2 contains points (8,5) and (2,4)