Download Determine Whether an Ordered Pair is a Solution

Math 35
2.2 "Graphing Linear Equations in Two Variables"
Goals:
* Determine whether an ordered pair is a solution of an equation.
* Find a solution of an equation in two variables.
* Graph linear equations by plotting points.
* Graph linear equations by …nding intercepts.
* Graph horizontal and vertical lines.
* Solve applied problems.
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
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:
De…nition:
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
Intermediate Algebra by Tussy and Gustafson
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 = 32 x
y
-4
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
Intermediate Algebra by Tussy and Gustafson
2.2
Graph Horizontal and Vertical Lines
Equations such as y = 3 and x =
- y=3
- x=
2 are linear equations, because they can be written in standard form Ax + By = C:
is equivalent to
:
2 is equivalent to
:
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
-2
4
x
-4
-2
2
-2
-4
4
x
-4
Applied Problems
We will see how linear equations can model real-life situations. We can use linear models to make observations about
what has occurred in the past and what might occur in the future.
Example 6:
(Solving applied problems)
A copy machine that was purchased for $6; 750 is expected to depreciate according to the straight-line depreciation equation
y=
950x + 6750 where y is the value of the copier after x years of use. When will the copier have no value?
Page: 3
Notes by Bibiana Lopez