Using Graphs and Tables

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Using Graphs and Tables to Solve Linear Systems
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Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up Use substitution to determine if (1, 2)
is an element of the solution set of the linear
equation.
no
yes
1. y 2x 1
2. y 3x 5
Write each equation in slope-intercept form.
4. 4y 3x 8
3. 2y 8x 6
y 4x 3
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Objectives
Solve systems of equations by using graphs and
tables. Classify systems of equations, and
determine the number of solutions.
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Vocabulary
system of equations linear system consistent
system inconsistent system independent
system dependent system
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A system of equations is a set of two or more
equations containing two or more variables. A
linear system is a system of equations containing
only linear equations. Recall that a line is an
infinite set of points that are solutions to a
linear equation. The solution of a system of
equations is the set of all points that satisfy
each equation.
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• On the graph of the system of two equations, the
  solution is the set of points where the lines
  intersect. A point is a solution to a system of
  equation if the x- and y-values of the point
  satisfy both equations.

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Example 1A Verifying Solutions of Linear Systems
Use substitution to determine if the given
ordered pair is an element of the solution set
for the system of equations.
Substitute 1 for x and 3 for y in each equation.
Because the point is a solution for both
equations, it is a solution of the system.
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Example 1B Verifying Solutions of Linear Systems
Use substitution to determine if the given
ordered pair is an element of the solution set
for the system of equations.
x 6 4y
(4, )
2x 8y 1
?
x
Because the point is not a solution for both
equations, it is not a solution of the system.
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Check It Out! Example 1a
Use substitution to determine if the given
ordered pair is an element of the solution set
for the system of equations.
Substitute 4 for x and 3 for y in each equation.
Because the point is a solution for both
equations, it is a solution of the system
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Check It Out! Example 1b
Use substitution to determine if the given
ordered pair is an element of the solution set
for the system of equations.
Substitute 5 for x and 3 for y in each equation.
Because the point is not a solution for both
equations, it is not a solution of the system.
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Recall that you can use graphs or tables to find
some of the solutions to a linear equation. You
can do the same to find solutions to linear
systems.
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Example 2A Solving Linear Systems by Using
Graphs and Tables
Use a graph and a table to solve the system.
Check your answer.
2x 3y 3
y 2 x
Solve each equation for y.
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Example 2A Continued
On the graph, the lines appear to intersect at
the ordered pair (3, 1)
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Example 2A Continued
y x 2

• Make a table of values for each equation.
  Notice that when x 3, the y-value for both
  equations is 1.

The solution to the system is (3, 1).
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Example 2B Solving Linear Systems by Using
Graphs and Tables
Use a graph and a table to solve the system.
Check your answer.
Solve each equation for y.
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Example 2B Continued
Use your graphing calculator to graph the
equations and make a table of values. The lines
appear to intersect at (3, 5). This is the
confirmed by the tables of values.
The solution to the system is (3, 5).
Check Substitute (3, 5) in the original
equations to verify the solution.
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Check It Out! Example 2a
Use a graph and a table to solve the system.
Check your answer.
Solve each equation for y.
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Check It Out! Example 2a Continued
On the graph, the lines appear to intersect at
the ordered pair (0, 3)
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Check It Out! Example 2a Continued
y 4x 3
y x 3
Make a table of values for each equation. Notice
that when x 0, the y-value for both equations
is 3.
The solution to the system is (0, 3).
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Check It Out! Example 2b
Use a graph and a table to solve the system.
Check your answer.
x y 8
2x y 4
Solve each equation for y.
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Check It Out! Example 2b Continued
On the graph, the lines appear to intersect at
the ordered pair (4, 4).
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Check It Out! Example 2b Continued
y 2x 4
y 8 x
Make a table of values for each equation. Notice
that when x 4, the y-value for both equations
is 4.
The solution to the system is (4, 4).
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Check It Out! Example 2c
Use a graph and a table to solve each system.
Check your answer.
y x 5
3x y 1
y x 5
Solve each equation for y.
y 3x 1
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Check It Out! Example 2c Continued
On the graph, the lines appear to intersect at
the ordered pair (1, 4).
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Check It Out! Example 2c Continued
y 3x 1
y x 5
Make a table of values for each equation. Notice
that when x 1, the y-value for both equations
is 4.
The solution to the system is (1, 4).
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The systems of equations in Example 2 have
exactly one solution. However, linear systems may
also have infinitely many or no solutions. A
consistent system is a set of equations or
inequalities that has at least one solution, and
an inconsistent system will have no solutions.
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• You can classify linear systems by comparing
  the slopes and y-intercepts of the equations. An
  independent system has equations with different
  slopes. A dependent system has equations with
  equal slopes and equal y-intercepts.

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Example 3A Classifying Linear System
Classify the system and determine the number of
solutions.
x 2y 6
3x 6y 18
The equations have the same slope and
y-intercept and are graphed as the same line.
Solve each equation for y.
The system is consistent and dependent with
infinitely many solutions.
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Example 3B Classifying Linear System
Classify the system and determine the number of
solutions.
4x y 1
y 1 4x
The equations have the same slope but different
y-intercepts and are graphed as parallel lines.
Solve each equation for y.
The system is inconsistent and has no solution.
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Example 3B Continued
Check A graph shows parallel lines.
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Check It Out! Example 3a
Classify the system and determine the number of
solutions.
7x y 11
3y 21x 33
The equations have the same slope and
y-intercept and are graphed as the same line.
Solve each equation for y.
The system is consistent and dependent with
infinitely many solutions.
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Check It Out! Example 3b
Classify each system and determine the number of
solutions.
x 4 y
5y 5x 35
The equations have the same slope but different
y-intercepts and are graphed as parallel lines.
Solve each equation for y.
The system is inconsistent with no solution.
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Example 4 Summer Sports Application
City Park Golf Course charges 20 to rent golf
clubs plus 55 per hour for golf cart rental. Sea
Vista Golf Course charges 35 to rent clubs plus
45 per hour to rent a cart. For what number of
hours is the cost of renting clubs and a cart the
same for each course?
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Example 4 Continued
Step 1 Write an equation for the cost of renting
clubs and a cart at each golf course.
Let x represent the number of hours and y
represent the total cost in dollars.
City Park Golf Course y 55x 20
Sea Vista Golf Course y 45x 35
Because the slopes are different, the system is
independent and has exactly one solution.
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Example 4 Continued
Step 2 Solve the system by using a table of
values.
y 55x 20
y 45x 35
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Check It Out! Example 4
Ravi is comparing the costs of long distance
calling cards. To use card A, it costs 0.50 to
connect and then 0.05 per minute. To use card B,
it costs 0.20 to connect and then 0.08 per
minute. For what number of minutes does it cost
the same amount to use each card for a single
call?
Step 1 Write an equation for the cost for each of
the different long distance calling cards.
Let x represent the number of minutes and y
represent the total cost in dollars.
Card B y 0.08x 0.20
Card A y 0.05x 0.50
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Check It Out! Example 4 Continued
Step 2 Solve the system by using a table of
values.
y 0.05x 0.50
y 0.08x 0.20
When x 10 , the y-values are both 1.00. The
cost of using the phone cards of 10 minutes is
1.00 for either cards. So the cost is the same
for each phone card at 10 minutes.