Download Determine Whether an Ordered Pair is a Solution of an Equation

Math 102
2.2 "Graphing Techniques: Linear Equations and Inequalities"
Objectives:
* Graph linear equations.
* Find the x and y intercepts for the graph of a linear equation.
* Graph linear inequalities.
In this section, we will discuss equations that contain two variables. These equations are used to describe algebraic
relationships between two quantities.
Determine Whether an Ordered Pair is a Solution of an Equation
De…nition:
Solution
kA solution of an equation in two variables is an ordered pair of real numbers that satis…es the equation.k
Note:
When the variables are x and y, the ordered pairs are of the form (x; y)
:
Example 1: (Checking for solutions)
Determine whether each ordered pair is a solution of the equation:
a) ( 4; 2)
2x
5y = 9
b) (2; 1)
Find a Solution of an Equation in Two Variables
Note:
To …nd a solution of an equation in two variables, we can select a number for one of the variables and …nd the
corresponding value of the other variable.
Example 2: (Finding solutions of an equation in two variables)
Complete the following table of solutions.
x
y=x 2
2
0
4
Graph Linear Equations by Plotting Points
Equations in two variables can be graphed in several ways. If an equation in x and y is solved for y, we can graph it
by selecting values for x and calculating the corresponding values of y:
Standard (General) Form of a Linear Equation:
A linear equation in two variables is an equation that can be written in the form
where A; B; and C are real numbers and A and B are not both zero.
Page: 1
Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
2.2
Strategy:
Graphing Linear Equations
Step 1: Solve for y
Step 2: Find three ordered pairs that satisfy the equation
Step 3: Plot the solutions on a rectangular coordinate system
Step 4: Draw a straight line passing through the points
Example 3: (Graphing linear equations)
Graph 4x + 3y = 6 by using the above strategy.
Step 1
Step 2
x
Step 3&4
y
y
(x; y)
4
2
-4
-2
2
-2
4
x
-4
Graph Linear Equations by Finding Intercepts
Plotting the x intercept and y intercept of a graph and drawing a line through them is called the intercept method
of graphing a line. It is useful when graphing linear equations in standard form.
De…nition:
Intercepts of a Line
The x intercept of a line is the point (a; 0) , where the line intersects the x
axis.
To …nd a, substitute 0 for y in the equation of the line and solve for x:
The y intercept of a line is the point (0; b) , where the line intersects the y
axis.
To …nd b, substitute 0 for x in the equation of the line and solve for y:
Example 4: (Graphing linear equations)
Graph the following linear equations by …nding the intercepts.
a) 3x
4y = 12
b) y =
y
-4
2
x
3
2
y
4
4
2
2
-2
2
-2
4
-4
x
-2
2
-2
-4
4
x
-4
Page: 2
Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
2.2
Graph Horizontal and Vertical Lines
Equations of Horizontal and Vertical Lines:
The equation
represents the horizontal line that intersects the y
The equation
represents the vertical line that intersects the x
The graph of the equation y = 0 is the x
Note:
axis at (0; b)
axis at (a; 0)
axis: Similarly, the graph of the equation x = 0 is the y
axis:
Example 5: (Graphing linear equations)
Graph the following equations.
a)
x=
3
b)
y
-4
y=4
y
4
4
2
2
-2
2
4
-2
-4
x
-2
2
4
x
-2
-4
-4
Graphing Linear Inequalities
De…nition:
Linear Inequalities
Linear inequalities in two variables are of the form
or
where A; B; and C represent real numbers: Combined linear equality and inequality statements are of the form
or
Example 6: (Graphing linear inequalities)
Graph the following linear inequalities.
a) x
2y > 4
-4
b) 2x + 3y
6
4
4
2
2
-2
2
4
-4
-2
2
-2
-2
-4
-4
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4
Notes by Bibiana Lopez