Cartesian Plane and Linear Equations in Two Variables

Cartesian Plane and
Linear Equations in
Two Variables
Math 021
• 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
• 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.
Examples – Plot each if the following ordered
pairs on the Cartesian Plane and name the
quadrant it lies in:
10
9
8
7
6
5
4
3
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9
10
a.
b.
c.
d.
e.
f.
A = (3, 4)
B = (-2, 1)
C = (7, -3)
D = (-4, -2)
E = (0, 5)
F = (-1, 0)
• 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.
• 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)
• 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)
Complete the table of values for each
equation:
•
y = 2x – 10
x
y
x + 3y = 9
x
4
0
-20
5
y
6
4
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.
Graph: 2x + y = 4
10
9
8
7
6
5
4
3
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9
10
Graph: y= 3x-1
10
9
8
7
6
5
4
3
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9
10
Graph: y= 2x
10
9
8
7
6
5
4
3
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9
10
Graph: 15= -5y + 3x
10
9
8
7
6
5
4
3
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9
10
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 xintercept is 0.
• A y-intercept crosses the y-axis.
The x-coordinate of any yintercept is 0.
Graph by Finding Intercepts:
3x – 2y = 12
10
9
8
7
6
5
4
3
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9
10
Graph by Finding Intercepts:
y= -2x + y
10
9
8
7
6
5
4
3
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9
10
Graph by Finding Intercepts:
4x + 3y = -12
10
9
8
7
6
5
4
3
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9
10
Graph by Finding Intercepts:
3x – 5y = -15
10
9
8
7
6
5
4
3
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9
10
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.
Graph: x = 4
10
9
8
7
6
5
4
3
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9
10
Graph: y= -2
10
9
8
7
6
5
4
3
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9
10
Graph: 3x = -15
10
9
8
7
6
5
4
3
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9
10
Graph: y + 3 = 4
10
9
8
7
6
5
4
3
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9
10
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)
• Find the slopes of the lines below:
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= -3
3y -2 = 4
10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
1
1
2
3
4
5
6
7
8
9
10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
-5
-6
-6
-7
-7
-8
-8
-9
-9
-10
-10
1
2
3
4
5
6
7
8
9
10
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 yintercept can be easily identified.
Examples – Find the slope and yintercept of each equation:
•a.
•b.
•c.
•d.
y = 3x – 2
4y = 5x + 8
4x + 2y = 7
5x – 7y = 11
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 ┴
Complete the following table:
Slope
a.

2
7
b.
5
c.
0
// Slope
┴ slope
Examples – Determine if each pair of lines is
parallel, perpendicular, or neither:
• a. 2y = 4x + 7
y – 2x = -3
• c. 3x + 4y = 3
4x + 5y = -1
b. 5x – 10y = 6
y = 2x + 7
d. Line 1 contains points (3,1) and (2,7)
Line 2 contains points (8,5) and (2,4)