3.1
Solving Linear Systems
by Graphing
Goals p Graph and solve systems of linear equations.
p Use linear systems to solve real-life problems.
Your Notes
VOCABULARY
System of two linear equations
Solution of a system of linear equations
Example 1
Checking Solutions of a Linear System
Check whether (a) (1, 4) and (b) (1, 2) are solutions of the
following system.
3x y 1
xy3
Solution
a. 3(
)(
(
)(
b. 3(
)(
(
)(
)
)
)
Because (1, 2)
(1, 4)
(1, 4)
54
Algebra 2 Notetaking Guide • Chapter 3
)
Equation 1
Equation 2
Equation 1
.
Equation 2
.
Equation 1
.
Equation 2
.
a solution of each equation and
, (1, 2)
a solution of the system and
.
Your Notes
Solving a System Graphically
Example 2
Solve the system.
2x y 5
x 3y 8
Equation 1
Equation 2
Solution
Begin by graphing both equations.
From the graph, the lines appear to
intersect at (
,
). You can check
this algebraically as follows.
2(
)
3(
y
5
3
1
3
)
The solution is (
,
1
1
1
3 x
).
Systems with Many or No Solutions
Example 3
Tell how many solutions the linear system has.
a. x 3y 1
x 3y 1
b. x 3y 1
3x 9y 3
Solution
a. Graph the linear system.
b. Graph the linear system.
y
3
y
3
3
1
1
1
1
1
3
3 x
The graphs of the equations
are two
lines.
Because the two lines have
,
the system has
.
1
1
1
3 x
The graphs of the equations
are the
.
So, each point on the line is
a solution and the system
has
.
Lesson 3.1 • Algebra 2 Notetaking Guide
55
Your Notes
NUMBER OF SOLUTIONS OF A LINEAR SYSTEM
The relationship between the graph of a linear system and
the system’s number of solutions is described below.
Graphical Interpretation
Algebraic Interpretation
The graph of the system is The system has
a pair of lines that intersect
in one point.
The graph of the system is
a single line.
The system has
The graph of the system is
a pair of parallel lines so
that there is no point of
intersection.
The system has
.
.
y
y
y
x
x
Exactly one solution
.
Infinitely many solutions
x
No solution
Checkpoint Complete the following exercises.
1. Solve the system. Check
your solution.
x 7y 29
9x y 5
56
Algebra 2 Notetaking Guide • Chapter 3
2. Tell how many solutions
the linear system has.
2
x y 2
3
4x 6y 12
Your Notes
Writing and Using a Linear System
Example 4
You are putting a fence around a rectangular yard and you
want the length to be twice the width. You will use 120 feet
of fencing. What are the dimensions of the fence?
Solution
Verbal
Model
Length 2(
)
2( Length ) 2( Width ) Labels
Equation 1
Equation 2
Algebraic l Model 2l 2w Length l
(feet)
Width w
(feet)
Length l
(feet)
Width w
(feet)
Perimeter (feet)
Equation 1
Equation 2
Graph the linear system. The lines
appear to intersect at the point
(
,
). Check this algebraically.
2(
)
2(
) 2(
l
60
40
)
20
The solution is (
,
), which
means that the fence has a length
of
feet and a width of
feet.
20
40
60
w
Checkpoint Complete the following exercise.
Homework
3. You decide instead to use 105 feet of fencing in
Example 4 and you want the length to be 4 times the
width. Write a linear system for this situation.
Lesson 3.1 • Algebra 2 Notetaking Guide
57
3.2
Solving Linear Systems
Algebraically
Goals p Use algebraic methods to solve linear systems.
p Use linear systems to model real-life situations.
Your Notes
THE SUBSTITUTION METHOD
Step 1 Solve one of the equations for one of its variables.
Step 2 Substitute the expression from
into the
other equation and solve for the other variable.
Step 3 Substitute the value from
equation from Step 1 and solve.
Example 1
into the revised
The Substitution Method
Solve the linear system using the substitution method.
x ⫺ 2y ⫽ 4
2x ⫺ 3y ⫽ 1
Equation 1
Equation 2
Solution
1. Solve Equation 1 for x.
x ⫺ 2y ⫽ 4
Write Equation 1.
x⫽
Revised Equation 1
2. Substitute this expression into Equation 2 and solve for y.
2x ⫺ 3y ⫽ 1
2(
) ⫺ 3y ⫽ 1
y⫽
Write Equation 2.
Substitute for x.
Solve for y.
3. Substitute the value of y into revised Equation 1 and
solve for x.
x⫽
Write revised Equation 1.
x⫽
Substitute for y.
x⫽
Simplify.
The solution is (
,
). Check the solution by
substituting back into the original equations.
58
Algebra 2 Notetaking Guide • Chapter 3
Your Notes
THE LINEAR COMBINATION METHOD
Step 1 Multiply one or both of the equations by a
to obtain coefficients that differ only in
for one of the variables.
Step 2 Add the revised equations from
. Combining
like terms will
one of the variables.
Solve for the remaining variable.
Step 3 Substitute the value obtained in
into either
of the original equations and solve for the other
variable.
Example 2
The Linear Combination Method
Solve the linear system using
2x ⫺ 3y ⫽ 1
the linear combination method. ⫺4x ⫹ 2y ⫽ 6
1. Multiply the first equation by
differ only in sign.
2x ⫺ 3y ⫽ 1
Equation 1
Equation 2
so that the x-coefficients
ⴛ
Equation 1
⫺4x ⫹ 2y ⫽ 6
⫺4x ⫹ 2y ⫽ 6
2. Add the revised equations
and solve for y.
Equation 2
y⫽
3. Substitute the value of y into one of the original
equations. Solve for x.
2x ⫺ 3y ⫽ 1
2x ⫺ 3(
2x ⫹
Write Equation 1.
)⫽1
Substitute for y.
⫽1
Simplify.
x⫽
The solution is
Solve for x.
.
Check You can check the solution
algebraically using the method shown
in Example 1. You can also use a
graphing calculator to check the
solution.
Intersection
X=-2.5
Y=-2
Lesson 3.2 • Algebra 2 Notetaking Guide
59
Your Notes
Checkpoint Complete the following exercises.
1. Solve the linear system
using the substitution
method.
x⫺y⫽3
⫺3x ⫹ 4y ⫽ 2
Example 3
2. Solve the linear system
using the linear
combination method.
5x ⫹ 6y ⫽ ⫺2
2x ⫹ 3y ⫽ 1
The Linear Combination Method
Solve the linear system using
the linear combination method.
2x ⫹ 5y ⫽ 6
⫺3x ⫺ 2y ⫽ 2
Equation 1
Equation 2
Solution
Multiply the first equation by
and the second equation
by
so that the coefficients of x differ only in sign.
2x ⫹ 5y ⫽ 6
ⴛ
⫺3x ⫺ 2y ⫽ 2
ⴛ
Add the revised equations
and solve for y.
y⫽
Substitute the value of y into one of the original equations.
Solve for x.
2x ⫹ 5y ⫽ 6
2x ⫹ 5(
)⫽6
Substitute for y.
x⫽
Solve for x.
The solution is (
graphically.
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Algebra 2 Notetaking Guide • Chapter 3
Write Equation 1.
,
). Check the solution algebraically or
Your Notes
Example 4
Linear Systems with Many or No Solutions
Solve the linear system.
a. 4x ⫺ 2y ⫽ 5
2x ⫺ y ⫽ 3
b. ⫺3x ⫹ 6y ⫽ 9
4x ⫺ 8y ⫽ ⫺12
Solution
a. Because the coefficient of y in the second equation is
, use substitution. Solve the second equation for y.
2x ⫺ y ⫽ 3
⫽y
Substitute the expression for y into the first equation.
4x ⫺ 2y ⫽ 5
4x ⫺ 2(
Write first equation.
)⫽5
Substitute for y.
⫽
Simplify.
Because the statement
is
.
, there
b. Because no coefficient is
, use the linear
combination method. Multiply the first equation by
the second equation by
.
⫺3x ⫹ 6y ⫽ 9
4x ⫺ 8y ⫽ ⫺12
and
ⴛ
ⴛ
Add the revised equations.
Because the equation
is
, there
.
Checkpoint Solve the linear system.
Homework
3. 2x ⫹ 7y ⫽ ⫺4
⫺6x ⫺ 21y ⫽ 12
4. 3x ⫺ 5y ⫽ 10
7x ⫺ 11y ⫽ 20
Lesson 3.2 • Algebra 2 Notetaking Guide
61
3.3
Graphing and Solving Systems
of Linear Inequalities
Goals p Graph a system of linear inequalities.
p Use systems of linear inequalities to solve real-life
problems.
Your Notes
VOCABULARY
System of linear inequalities in two variables
Solution of a system of linear inequalities
Graph of a system of linear inequalities
GRAPHING A SYSTEM OF LINEAR INEQUALITIES
To graph a system of linear inequalities, do the following for
each inequality in the system:
p Graph the line that corresponds to the inequality. Use a
line for an inequality with < or > and a
line for an inequality with ≤ or ≥.
p Lightly shade the half-plane that is the graph of the
inequality. Colored pencils may help you distinguish the
different half-planes.
The graph of the system is the region common to all of the
half-planes. If you used colored pencils, it is the region that
has been shaded with
color.
62
Algebra 2 Notetaking Guide • Chapter 3
Your Notes
Example 1
Graphing a System of Two Inequalities
y≥x⫺1
y < ⫺2x ⫹ 1
Graph the system.
Inequality 1
Inequality 2
Begin by graphing each linear inequality. Use different shades
for each half-plane.
y
Shade the half-plane
on and
y ⴝ x ⴚ 1.
3
The graph of the system
is the intersection of
the two shaded regions.
1
⫺1
⫺1
⫺3
1
3
x
Shade the half-plane
of
to the
y ⴝ ⴚ2x ⴙ 1.
⫺3
Example 2
Graphing a System of Three Inequalities
x≥0
y≥0
2y ⫹ 3x < 7
Graph the system.
Inequality 1
Inequality 2
Inequality 3
The graph of the system is a
region.
The inequality 2y ⴙ 3x < 7
implies that the region is
the line
2y ⴙ 3x ⴝ 7.
y
The inequality x ≥ 0
implies that the region
is on and to the
of the y-axis.
5
3
1
⫺1
⫺1
1
3
5 x
The inequality y ≥ 0
implies that the region is
on and
the
x-axis.
Checkpoint Complete the following exercise.
1. Graph the system.
x≥1
y≥2
y≤5⫺x
y
5
3
1
⫺1
⫺1
1
3
5 x
Lesson 3.3 • Algebra 2 Notetaking Guide
63
Your Notes
Example 3
Writing and Using a System of Inequalities
You are selling $5 adult tickets and $3 youth tickets for a
school play. You want to sell at least $600 worth of tickets.
a. Write and graph a system of linear inequalities that
describes the information given above.
b. You sell 80 adult tickets and 100 youth tickets. Did you
reach your sales goal?
Solution
a. Let y represent the number of youth tickets sold. Let x
represent the number of adult tickets sold. From the given
information, you can write the following system of
inequalities.
x≥
Must sell at least
adult tickets.
y≥
Must sell at least
youth tickets.
Total sales must be at least $
Graph the system of inequalities.
.
School Play Target Sales
y
Youth Tickets Sold
b. From the graph you can see that
(80, 100)
in the solution
region, so selling 80 adult
tickets and 100 youth tickets
reach your sales goal.
150
100
50
0
0
40
80
120
Adult Tickets Sold
Checkpoint Complete the following exercise.
School Play Target Sales
y
Youth Tickets Sold
Homework
2. Assume that in Example 3,
the auditorium can only hold
up to 200 people. Write and
graph a system of linear
inequalities with this addition.
150
100
50
0
0
60
120
180
Adult Tickets Sold
64
Algebra 2 Notetaking Guide • Chapter 3
x
x
3.4
Linear Programming
Goals p Solve linear programming problems.
p Use linear programming to solve real-life problems.
Your Notes
VOCABULARY
Optimization
Linear programming
Objective function
Constraints
Feasible region
OPTIMAL SOLUTION OF A LINEAR PROGRAMMING
PROBLEM
If an objective function has a maximum or a minimum
value, then it must occur at a
of the feasible
region. Moreover, the objective function will have both a
maximum and a minimum value if the feasible region is
.
y
y
x
Bounded region
x
Unbounded region
Lesson 3.4 • Algebra 2 Notetaking Guide
65
Your Notes
Example 1
Solving a Linear Programming Problem
Find the minimum value and the maximum value of
C 2y 3x
Objective function
subject to the following constraints.
x 2
y 1
yx 3
Constraints
Solution
The feasible region determined by the
constraints is shown. The three vertices
are (4, 1), (2, 1), and (2, 5). To find
the minimum and maximum values of
C, evaluate C 2y 3x at each of the
three vertices.
5
y (2, 5)
3
1
3
1
(4, 1)
1
At (4, 1): C At (2, 1): C At (2, 5): C The minimum value of C is
. It occurs when x and y . The maximum value of C is
. It occurs
when x and y .
Checkpoint Complete the following exercise.
1. Find the minimum value and maximum value of
C x 2y subject to the following constraints.
x
y
y
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Algebra 2 Notetaking Guide • Chapter 3
1
x1
7x
x
(2, 1)
M2ng_nt.03
2/18/05
9:06 AM
Page 67
Your Notes
A Region that is Unbounded
Example 2
Find the minimum value and the maximum value of
C y 2x
Objective function
subject to the following constraints.
x
y
2y
3y
0
0
3x 8
x 5
Constraints
Solution
The feasible region determined by the
constraints is shown. The three vertices
are (0, 4), (2, 1), and (5, 0). Evaluate
C y 2x at each of the vertices.
At (0, 4): C 4 2(0) 4
5
y
(0, 4)
3
1
1
(5, 0)
(2, 1)
1
3
5 x
At (2, 1): C 1 2(2) 5
At (5, 0): C 0 2(5) 10
If you evaluate other points in the feasible region, you will see
that as the points get farther from the origin, the value of the
objective function increases without bound. So, the
objective function has no maximum value. The value of the
objective function is always at least 4 , so the minimum
value is 4 .
Checkpoint Complete the following exercise.
2. Find the minimum value and the maximum value of
C 3x y subject to the following constraints.
x 4
x 0
y 2x
3y x
9
6
minimum: 9, maximum: none
Lesson 3.4 • Algebra 2 Notetaking Guide
67
Your Notes
Finding Maximum Profit
Example 3
You are making fruit baskets with oranges, bananas, and
apples. The table gives the amount of fruit required for
the two arrangements. Each day you have 240 oranges,
270 bananas, and 320 apples. Arrangement A earns a profit
of $10 per basket and Arrangement B earns $8 per basket.
How many of each fruit basket should you make per day to
maximize your profit?
Fruit
Arrangement A
Arrangement B
Oranges
4
6
Bananas
6
6
Apples
8
6
Solution
Write an objective function. Let a and b
represent the number of each type of
fruit basket made. Because you want to
maximize the profit P, the objective
function is: P .
50
b
(0, 40)
(15, 30)
30
(25, 20)
10
10
(40, 0)
10
30
50 a
Write the constraints in terms of a
and b.
a
b
Oranges: up to
a
b
Bananas: up to
a
b
Apples: up to
a
Amount cannot be
.
b
Amount cannot be
.
Calculate the profit at each vertex of the feasible region.
At (0, 40):
P
At (15, 30): P At (25, 20): P Homework
At (40, 0):
P
Maximum profit is obtained by making
baskets and
Arrangement B baskets.
68
Algebra 2 Notetaking Guide • Chapter 3
Arrangement A
3.5
Graphing Linear Equations
in Three Variables
Goals p Graph linear equations in three variables and evaluate
linear functions of two variables.
p Use functions of two variables to model real-life
situations.
Your Notes
VOCABULARY
Three-dimensional coordinate system
z-axis
Ordered triple
Octants
Linear equation in three variables
Function of two variables
Example 1
Plotting Points in Three Dimensions
Plot (ⴚ4, ⴚ1, 3) in a three-dimensional
coordinate system.
To plot (⫺4, ⫺1, 3), find the point
(
,
) in the xy-plane. The
point (⫺4, ⫺1, 3) lies 3 units
z
y
.
x
Lesson 3.5 • Algebra 2 Notetaking Guide
69
Your Notes
Graphing a Linear Equation in Three Variables
Example 2
Sketch the graph of 2x ⴚ 2y ⴙ 3z ⴝ 6.
Find points at which the graph
z
intersects the axes. Let x ⫽ 0 and
y ⫽ 0, and solve for z to get z ⫽
.
y
This tells you that the z-intercept is
, so plot ( , ,
). In a similar
x
way, you can find that the x-intercept
is
and the y-intercept is
. After plotting the intercepts,
connect the points with lines to form the triangular region of
the plane.
Evaluating a Function of Two Variables
Example 3
Write the linear equation 2x ⴚ 2y ⴙ 3z ⴝ 6 as a function
of x and y. Then evaluate the function when x ⴝ 1 and
y ⴝ 1. Interpret the result geometrically.
2x ⫺ 2y ⫹ 3z ⫽ 6
Write original equation.
3z ⫽
Isolate z-term.
z⫽
Solve for z.
f(x, y) ⫽
Replace z with f(x, y).
⫽
f(1, 1) ⫽
graph of f contains the point (
,
. This tells you that the
,
).
Checkpoint Complete the following exercise.
1. Graph 3x ⫹ 4y ⫺ 4z ⫽ 12. Then write the equation as a
function of x and y and evaluate f(0, 2).
z
y
x
70
Algebra 2 Notetaking Guide • Chapter 3
Your Notes
Example 4
Modeling a Real-Life Situation
You are painting the inside of your house and plan to use
two colors of paint: white and blue. White costs $12 per
gallon and blue costs $15 per gallon. You also buy supplies
totaling $40.
a. Write a model for the total amount you spend as a
function of the number of gallons of white and blue paint
you use.
b. Evaluate the model for 3 gallons of white paint and
2 gallons of blue paint.
Solution
a. Verbal
Model
Labels
Total ⫽
White
p
cost
paint cost
⫹
Blue paint p
cost
⫹
Total Cost ⫽ C
(dollars)
White paint cost ⫽
(dollars per gallon)
White paint amount ⫽ w
(gallons)
Blue paint cost ⫽
(dollars per gallon)
Blue paint amount ⫽ b
(gallons)
Supplies ⫽
(dollars)
Supplies
Algebraic C ⫽
Model
b. Substitute w ⫽
total cost.
Homework
and b ⫽
into the function to find the
C⫽
Write original function.
⫽
Substitute for w and b.
⫽
Simplify.
The total cost is $
.
Lesson 3.5 • Algebra 2 Notetaking Guide
71
3.6
Solving Systems of Linear
Equations in Three Variables
Goals p Solve systems of linear equations in three variables.
p Use linear systems in three variables to model real-life
situations.
Your Notes
VOCABULARY
System of three linear equations
Solution of a system of three linear equations
THE LINEAR COMBINATION METHOD
(3-VARIABLE SYSTEMS)
Step 1 Use the linear combination method to rewrite the
linear system in three variables as a linear system in
variables.
Step 2 Solve the new linear system for both of its variables.
Step 3 Substitute the values found in
into one of
the original equations and solve for the remaining
variable.
Note: If you obtain a
equation, such as 0 ⫽ 1, in any
of the steps, then the system has
. If you
obtain an
, such as 0 ⫽ 0, then the system has
.
72
Algebra 2 Notetaking Guide • Chapter 3
Your Notes
Example 1
Using the Linear Combination Method
Solve the system.
4x ⫹ 2y ⫺ z ⫽ 1
x ⫺ 3y ⫹ 2z ⫽ 14
⫺x ⫹ y ⫹ z ⫽ ⫺1
Equation 1
Equation 2
Equation 3
1. Eliminate a variable in two of the original equations.
4x ⫹ 2y ⫺ z ⫽ 1
Add
times the third
equation to the first.
New Equation 1
x ⫺ 3y ⫹ 2z ⫽ 14
⫺x ⫹ y ⫹ z ⫽ ⫺1
Add the second and
third equations.
New Equation 2
2. Solve the new system of linear equations.
Add
times new Equation 2
to new Equation 1.
y⫽
Solve for y.
z⫽
Substitute y in new Equation 1 or 2.
3. Substitute y and z into an original equation. Solve for x.
⫺x ⫹ y ⫹ z ⫽ ⫺1
Equation 3
Substitute for y and for z.
x⫽
Solve for x.
The solution is the ordered triple ( ,
,
solution in each of the original equations.
). Check this
Checkpoint Complete the following exercise.
1. Solve the system
x ⫹ 3y ⫺ 4z ⫽ ⫺3
⫺2x ⫹ 2y ⫹ z ⫽ 0
3x ⫺ 2y ⫹ 2z ⫽ 8
Lesson 3.6 • Algebra 2 Notetaking Guide
73
Your Notes
Example 2
Solving a System with Many Solutions
Solve the system.
x ⫹ 2y ⫹ 2z ⫽ 4
2x ⫺ 2y ⫹ 4z ⫽ 8
2x ⫹ 2y ⫹ 4z ⫽ 8
Equation 1
Equation 2
Equation 3
Rewrite the linear system in three variables as a linear
system in two variables.
x ⫹ 2y ⫹ 2z ⫽ 4
2x ⫺ 2y ⫹ 4z ⫽ 8
Add the first equation
to the second.
New Equation 1
2x ⫺ 2y ⫹ 4z ⫽ 8
2x ⫹ 2y ⫹ 4z ⫽ 8
Add the second equation
to the third.
New Equation 2
The result is a system of linear equations in two variables.
New Equation 1
New Equation 2
Solve the new system by adding
to
times the first equation
times the second equation. This produces the
. So, the system has
.
Describe the solution. Rewrite new Equation 1 to get
x⫽
. Substituting this into original equation 1
produces y ⫽
. So, any ordered triple of the form
(
, , ) is a solution of the system. For instance,
( , , 2), ( , , 1), and ( , , 0) are all solutions.
Checkpoint Solve the system.
2. x ⫹ y ⫺ 2z ⫽ 3
2x ⫹ 2y ⫺ 4z ⫽ 1
⫺x ⫺ 3y ⫹ z ⫽ 5
Homework
74
Algebra 2 Notetaking Guide • Chapter 3
3. x ⫹ 2y ⫺ 4z ⫽ 7
⫺x ⫺ y ⫹ 2z ⫽ ⫺4
x ⫹ 4y ⫺ 8z ⫽ 13