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

Math 35 "Fall ’08"
2.2 "Graphing Linear Equations in Two Variables"
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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.
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In this section, we will discuss equations that contain two variables.
This 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)
b)
2x
5y = 9
(2; 1)
Find a Solution of an Equation in Two Variables
To …nd a solution of an equation in two variables, we can select a number for one of the variables and …nd
Note:
the corresponding value of the other variable.
Example 2: (Finding solutions of an equation in two variables)
Complete the following table of solutions:
y=x
x
2
y
2
0
4
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Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
2.2
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.
Examples of linear equations
Strategy:
Nonexamples:
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:
y
4
2
-4
-2
2
-2
4
x
-4
Graph Linear Equations by Finding Intercepts
Plotting the x
intercepts and y
intercepts 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.
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Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
De…nition:
2.2
Intercepts of a Line
The
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:
The
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:
Example 4: (Graphing linear equations)
Graph 3x
4y = 12 equation by …nding the intercepts:
y
4
2
-4
-2
2
4
-2
x
-4
Example 5: (Graphing linear equations)
Graph y = 23 x
2 equation by …nding the intercepts:
y
4
2
-4
-2
2
4
-2
x
-4
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
.
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Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
Equations;
2.2
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 6: (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 7:
(Solve 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?
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Bibiana Lopez