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LINEAR EQUATIONS IN TWO VARIABLES
§ 4.1
Graphing Linear Equations in Two Variables
The Rectangular Coordinate System
In the rectangular coordinate system, the horizontal number line is the x-axis.
The vertical number line is the y-axis. The point of intersection of these axes is
their zero points, called the origin. The axes divide the plane into 4 quarters,
called quadrants.
y-axis
2nd quadrant
1st quadrant
x-axis
3rd quadrant
4th quadrant
The Rectangular Coordinate System
Each point in the rectangular coordinate system corresponds to an ordered pair of
real numbers (x,y). Note the word “ordered” because order matters. The first
number in each pair, called the x-coordinate, denotes the distance and direction
from the origin along the x-axis. The second number, the y-coordinate, denotes
vertical distance and direction along a line parallel to the y-axis or along the y-axis
Itself.
Point
Movement from origin
(3.5)
3 units right and 5 units up
(4,-3)
4 units right and 3 units down
(-2,-7)
2 units left and 7 units down
(0,0)
0 right or left and 0 up or down
In plotting points,
we move across first
(either left or right), and
then move either up or
down, always starting
from the origin.
Plotting Points
EXAMPLE
Plot the points (3,2) and (-2,-4).
SOLUTION
 (3,2)
(-2,-4) 
The Graph of an Equation
The graph of an equation in two variables is the set of
points whose coordinates satisfy the equation.
An ordered pair of real numbers (x,y)
is said to satisfy the equation when substitution of the x and y coordinates
into the equation makes it a true statement.
For example, in the equation
y = 2x + 6, the ordered pair (1,8) is a solution. When we substitute this
point the sentence reads 8 = 8, which is true.
The ordered pair (2,3) is not a solution. When we substitute this point, the
sentence reads 3 = 10, which is not true.
The Graph of an Equation
The graph of an equation in two variables is the set of
points whose coordinates satisfy the equation.
An ordered pair of real numbers (x,y)
is said to satisfy the equation when substitution of the x and y coordinates
into the equation makes it a true statement.
For example, in the equation
y = 2x + 6, the ordered pair (1,8) is a solution. When we substitute this
point the sentence reads 8 = 8, which is true.
The ordered pair (2,3) is not a solution. When we substitute this point, the
sentence reads 3 = 10, which is not true.
Solution of an Equation in Two Variables
Example:
Given the equation 2x + 3y = 18, determine if the
ordered pair (3, 4) is a solution to the equation.
We substitute 3 in for x and 4 in for y.
2(3) + 3 (4) ? 18
6 + 12 ? 18
18 = 18 True.
Therefore, the ordered pair (3, 4) is a solution to the
equation 2x + 3y = 18.
Finding Solutions of an Equation
Find five solutions to the equation y = 3x +
1.
Start by choosing some x values and then computing
the corresponding y values.
If x = -2, y = 3(-2) + 1 = -5.
5)
Ordered pair (-2, -
If x = -1, y = 3(-1) + 1 = -2.
-2)
Ordered pair ( -1,
If x =0, y = 3(0) + 1 = 1.
Ordered pair (0, 1)
If x =1, y = 3(1) + 1 =4.
Ordered pair (1, 4)
If x =2, y = 3(2) + 1 =7.
Ordered pair (2, 7)
Graph of the Equation
Plot the five ordered pairs to obtain the graph of y =
3x + 1
(2,7)
(1,4)
(0,1)
(-1,-2)
(-2,-5)
Blitzer, Introductory Algebra, 5e – Slide #10 Section 4.1
Graphing an Equation
EXAMPLE
Graphs in the rectangular coordinate system can also be used to tell a
story. Try to select the graph that best illustrates the story of the
population of the in INDIA.
Years
SOLUTION
Graph (c)
Population
(c)
Population
(b)
Population
(a)
Years
Years
Linear Equations
Definition of a Linear Equation
A linear equation in one variable x is an
equation that can be written in the form
ax + b = 0, where a and b are real numbers
and a is not equal to 0.
An example of a linear equation in x is 4x + 2 = 6.
Linear equations in x are first degree equations in
the variable x.
C
y
This is the graph of the
equation 2x + 3y = 12.
(0,4)
(6,0)
-2
x
2
The point (0,4) is the y-intercept.
The point (6,0) is the x-intercept.
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