Download Chapter 3: Systems of Linear Equations

 3.1 Graphically Solving Systems of Two Equations (Page 1 of 24) 3.1 Graphically Solving Systems of Two Equations
Definitions
The plot of all points that satisfy an equation forms the graph of the
equation. A system of linear equations is two or more linear equations
considered together. The system in example 1 is referred to as a system of
two linear equations in two variables. An ordered pair (a, b) that
satisfies all equations in a system of equations is a solution to the system
of equations. The set of all solutions to a system of equations makes up
the solution set to the system.
Example 1
Consider the system of linear equations:
y = 2x + 1
y = −3x + 6
1. The sketch the graph of each equation is shown. What is the point of
intersection in the two graphs?
2. Is there a point that satisfies both equations?
3. What is the relationship between the point(s) of intersection of the two
lines and the solution to the system of equations.
3.1 Graphically Solving Systems of Two Equations (Page 2 of 24) Example 2
Solve the system of equations by graphing.
y = 2x + 4
y = −x + 1
y
5
-5
5
x
-5
Example 3
Graph the system of equations. Use your calculators to estimate the
solution to the nearest hundredth.
1
5
5
x+ y=
y
2
4
2
y = 3x − 4
5
-5
5
-5
x
3.1 Graphically Solving Systems of Two Equations (Page 3 of 24) Example 4
Solve an Inconsistent System
Solve the system of equations by graphing.
y = 2x + 1
y = 2x − 5
y
5
-5
x
5
-5
Distinct Parallel lines form an Inconsistent System
Distinct lines with the same slope are parallel and do not intersect. In this
case the solution set is empty and the system of equations is called an
inconsistent system.
Example 5
Solve a Dependent System
Solve the system of equations by graphing.
y = 2x + 4
6x − 3y = −12
y
5
-5
5
-5
Non-Distinct lines form a Dependent System
Two lines with the same slope and same y-intercept are not distinct and
have infinitely many points of intersection. In this case the solution set is
every point on the line and the system of equations is called a dependent
system.
x
3.1 Graphically Solving Systems of Two Equations (Page 4 of 24) The Solution Set to a Linear System of Two Equations in Two
Variables has Three Possible Outcomes . . .
1. One Solution
See Figure 1. The lines
intersect at one point and the solution set
contains only one point. The solution to a
consistent system is expressed as an ordered pair
(a, b) .
y
x
Figure 1 A Consistent
System with one solution.
2. Inconsistent System See Figure 2. The lines
are distinct and parallel, and never intersect.
Therefore, there is no solution. A linear system
whose solution set is empty is called an
inconsistent system. The Elimination and
Substitution methods produce a false statement,
such as 4 = 0.
3. Dependent System
See Figure 3. The
lines lie on top of each other, are not distinct,
and have an infinite number of points of
intersection – every point on the line. Therefore,
the system of equations has infinitely many
solutions. This is called a dependent system. The
Elimination and Substitution methods produce a
true statement, such as 3 = 3.
y
x
Figure 2 An Inconsistent
System has no solutions.
y
x
Figure 3 A Dependent
System has an infinite
number of solutions.
3.1 Graphically Solving Systems of Two Equations (Page 5 of 24) Example 6
Some values for functions f and g are given in the
table.
1. What happens to the value of f each time x is
increased by one?
What is the slope of the linear function f?
What is the y-intercept of f?
x
0
1
2
3
4
5
6
7
f (x)
30
27
24
21
18
15
12
9
g(x)
2
7
12
17
22
27
32
37
Find the linear function f.
2. What happens to the value of g each time x is increased by one?
What is the slope of the linear function g?
What is the y-intercept of g?
Find the linear function g.
3. Use the Intersect function on your calculator to find the solution to the
system of equations given by f and g.
3.1 Graphically Solving Systems of Two Equations (Page 6 of 24) Example 7
The life expectancies for men and women in
the United States are given in the table. Let
M (t) and W (t) represent the life
expectancy of men and women,
respectively, for t years since 1980.
1. Verify the regression equations are
W (t) = 0.114t + 77.47 and
M (t) = 0.204t + 69.90 .
2. Find the point of intersection. Explain
its meaning in this application.
U.S. Life Expectancies of
Women and Men
Year of Women Men
Birth (years) (years)
1980
77.4
70.0
1985
78.2
71.1
1990
78.8
71.8
1995
78.9
72.5
2000
79.5
74.1
2006
80.7
75.4
3. What is the slope of each equation? What does it mean in the context
of this application?
y
Example 8
Write a system of equations that has
(−2, − 3) as its only solution.
5
-5
5
-5
x
3.2 Substitution and Elimination Methods (Page 7 of 24) 3.2 Solving a System of Equations by the Substitution and
Elimination Methods
What is the solution to the system of
equations shown? Notice how
difficult it is to read an accurate
solution when the components of the
ordered pair are non-integer values.
In section 3.1 we solved a system of
two equations with two variables by
graphing the two equations and
reading the point(s) of intersection as
the solution. In this section we will
learn two algebraic (symbolic)
methods to solve a system of two
equations with two variables: The
Substitution Method and the
Elimination Method.
y
4
2
x
-4
2
4
-2
-4
The Substitution Method
1. Solve one of the equations for one of the variables.
2. Substitute the equation from step 1 into the other equation and solve for
the only remaining variable.
3. Substitute the result from step 2 into the equation from step 1 to find the
value of the other variable.
4. Write the solution as an ordered pair.
Example 1
Solve by substitution −x + y = −1
3x + 2y = 13
Example 2
3.2 Substitution and Elimination Methods (Page 8 of 24) Solve using substitution
y = 11− 3x
y = 2x + 1
The Elimination Method
1. Rewrite one or both equations so that the coefficients on either variable
are opposites.
2. Add the two equations from step 1 together, eliminating one of the
variables. Solve for the value of the only variable present.
3. Use the result from step 2 and any equation to solve for the value of the
remaining unknown variable.
4. Write the solution as an ordered pair.
Example 3
Solve using elimination
4x − 5y = 3
3x + 5y = 11
Example 4
Solve using elimination
Example 5
Solve using elimination
3.2 Substitution and Elimination Methods (Page 9 of 24) 3x + 2y = 18
6x − 5y = 9
4x − 3y = −3
5x + 2y = 25
3.2 Substitution and Elimination Methods (Page 10 of 24) Example 6
Solve substitution or elimination.
y = 2x + 1
y = 2x + 3
Identifying an Inconsistent System of Two Equations
If the result of applying substitution or elimination is a false statement
(such as 1 = 3 ), then the system is inconsistent and the solution set is
empty. This occurs when the equations are two distinct lines with the same
slope (and different y-intercepts).
Example 7
Solve substitution or elimination.
y = 2x + 1
− 4y + 8x = − 4
Identifying a Dependent System of Two Equations
If the result of applying substitution or elimination is a true, but worthless,
statement (such as −4 = −4 ), then the system is dependent and infinitely
many solutions. This occurs when the lines are not distinct – the lines have
the same slope and same y-intercept.
3.2 Substitution and Elimination Methods (Page 11 of 24) The Solution Set to a Linear System of Two Equations in Two
Variables has Three Possible Outcomes . . .
1. One Solution
See Figure 1. The lines
intersect at one point and the solution set
contains only one point. The solution to a
consistent system is expressed as an ordered pair
(a, b) .
y
x
Figure 1 A Consistent
System with one solution.
2. Inconsistent System See Figure 2. The lines
are distinct and parallel, and never intersect.
Therefore, there is no solution. A linear system
whose solution set is empty is called an
inconsistent system. The Elimination and
Substitution methods produce a false statement,
such as 4 = 0.
3. Dependent System
See Figure 3. The
lines lie on top of each other, are not distinct,
and have an infinite number of points of
intersection – every point on the line. Therefore,
the system of equations has infinitely many
solutions. This is called a dependent system. The
Elimination and Substitution methods produce a
true statement, such as 3 = 3.
y
x
Figure 2 An Inconsistent
System has no solutions.
y
x
Figure 3 A Dependent
System has an infinite
number of solutions.
3.3 Using Systems to Model Data (Page 12 of 24)
3.3 Using Systems of Equations to Model Data
Example 1
Let M (t) and W (t) represent
the world record times of men
and women, respectively, for
the 400-meter run at t years
since 1900 (see table).
1. Verify the regression
equations are
W (t) = −0.274t + 70.454
and
M (t) = −0.053t + 48.079
World Record 400-Meter Run Times
Women
Men
Year Record Time Year Record Time
(seconds)
(seconds)
1957
57.0
1900
47.8
1959
53.4
1916
47.4
1962
51.9
1928
47.0
1969
51.7
1932
46.2
1972
51.0
1941
46.0
1976
49.29
1950
45.8
1979
48.60
1960
44.9
1983
47.99
1968
43.86
1985
47.60
1988
43.29
1999
43.18
2. Symbolically (i.e. show
your work algebraically) predict when the 400-meter run record time of
men and women will be equal. What will that record time be? Verify
your result on a graphing calculator using the INTERSECT function.
Example 2
In 2009, the price of a 2008 Cadillac De Ville was $26,440, and the price
for a 2008 Acura RSX was $18,243. The De Ville depreciates by $3548
per year, and the RSX depreciates by $1333 per year. When will the cars
have the same value? What will the value be? Show your work (i.e. do not
use your calculators).
3.4 Value, Interest, and Mixture Problems (Page 13 of 24) 3.4 VALUE, INTEREST AND MIXTURE PROBLEMS
Five-Step Problem Solving Method
Step 1
Define each variable.
Step 2
Write a system of two equations.
Step 3
Solve the system.
Step 4
Describe / Explain the result.
Step 5
Check your solution.
Total Value Formula
If n objects have a unit value v, then their total value T is given by
T = vn
i.e.
The total value of n nickels is 5n
The total value of n dimes is 10n
The total value of n quarters is 25n
Example 1
A music store charges $5 for a six-string pack of electric-guitar strings and
$20 for a four-string pack of electric-bass strings. If the store sells 35
packs of strings for a total of $295, how many packs of each type of string
were sold?
3.4 Value, Interest, and Mixture Problems (Page 14 of 24) Example 2
The American Analog Set will play at a auditorium that has 400 balcony
seats and 1600 main-level seats. If tickets for balcony seats cost $15 less
than the main-level seats, what should the price be for each type of ticket so
that the total revenue from a sellout performance will be $70,000?
3.4 Value, Interest, and Mixture Problems (Page 15 of 24) Example 3
A 10,000-seat amphitheater will sell general-seat tickets at $45 and
reserved-seat tickets at $65 for Foo Fighters concert. Let x and y be the
number of tickets that will sell for $45 and $65, respectively. Assume that
the show will sell out.
1. Let T = f (x) be the total revenue (in dollars) from selling the tickets.
Find the equation for f.
2. Use a graphing calculator to sketch the graph of f for 0 ≤ x ≤ 10, 000 .
What is the slope and what does it mean in this situation?
3. Find f (8500) and explain its meaning in this situation.
4. Find f (11, 000) and explain its meaning in this situation.
5. The total cost of the production is $350,000. How many of each type of
ticket must be sold to make a profit of $150,000?
3.4 Value, Interest, and Mixture Problems (Page 16 of 24) Simple Interest Problems
If P is the principal, r is the simple annual interest
rate, and I is the annual interest earned, then
I = P⋅r .
I = P⋅r
Example 4
How much interest will a person earn by investing $3200 in an account at a
4% simple annual interest for one year?
Example 5
A person plans to invest twice as much money in an Elfun Trust account at
2.7% annual interest than in a Vanguard Growth account at 5.5% annual
interest. Both interest rates are 5-year averages. How much will the person
have to invest in each account to earn a total of $218 in one year?
3.4 Value, Interest, and Mixture Problems (Page 17 of 24) Example 6
A person plans to invest a total of $6000 in Gabelli ABC mutual fund that
has a 3-year average annual interest of 6% and in a Presidential Bank
Internet CD accou8nt at 2.25% annual interest. Let x and y be the money
(in dollars) invested in the mutual fund and CD, respectively.
1. Let I = f (x) be the total interest (in dollars) earned from investing the
$6000 for one year. Find an equation for f.
2. Use a graphing calculator to draw a graph of f for 0 ≤ x ≤ 6000 . What
is the slope of f and what does it mean in this situation?
3. Use a graphing calculator to create a table of values for f. Explain how
such a table can be used to help a person decide how much money to
invest in each account.
4. How much money should be invested in each account to earn $300 in
one year?
3.4 Value, Interest, and Mixture Problems (Page 18 of 24) Mixture Problems
Let c be the percent concentration (in decimal form) of a substance in a
solution of volume V. Then the quantity Q of the pure substance is given by
c ⋅V = Q
e.g.
How many ounces of pure lime juice are in
100 oz of a 20% lime solution?
Example 7
A chemist needs 5 quarts of a 17% acid solution, but he has only a 15%
acid solution and a 25% acid solution. How many quarts of the 15% acid
solution should he mix with the 25% acid solution to make 5 quarts of a
17% acid solution?
Example 8
A chemist needs 8 cups of a 15% alcohol solution but has only a 20%
alcohol solution. How much water and 20% alcohol solution should she
mix to form the desired 8 cups of 15% solution?
3.5 Linear Inequalities in One Variable (Page 19 of 24) 3.5 Linear Inequalities in One Variable
A number satisfies an inequality in one variable if substituting the number
for the variable results in a true statement. Such numbers are called
solutions to the inequality. The solution set to an inequality is the set of
all solutions to the inequality. To solve an inequality means to find all the
solutions to the inequality. Two inequalities are said to be equivalent if
they have the same solutions set.
Some example of linear inequalities in one variable follow:
4 − x < 3 , 3x + 2 ≤ 5 , −8(2 − 3x) > 7 ,
−3
x≥5
7
Example 1
1. Is -3 a solution to 8 − 2x ≤ 7 ?
2. Does 6 satisfy the inequality −3(x − 2) < 6 ?
Addition / Subtraction Property of Inequalities
Let c be any real number or expression.
If a < b , then a + c < b + c .
In words, adding the same number or expression to both sides of an
inequality results in an equivalent inequality. Since subtraction is simply
addition of the opposite, subtracting
c
the same number of expression from
c
both sides of an inequality results in
a
b
a+c
b+c
an equivalent inequality.
3.5 Linear Inequalities in One Variable (Page 20 of 24) Multiplication / Division Property of Inequalities
1.
For a positive number c:
If a < b , then ac < bc
and
e.g. c = 3
2<5
If a > b , then ac > bc
e.g. c = 3
−5 > −12
2(3) < 5(3)
6 < 15
−5(3) > −12(3)
−15 > −36
In words this says multiplying both sides of an inequality by the same
positive number results in an equivalent inequality. Similarly, since
division is simply multiplication by the reciprocal, dividing both sides of
an inequality by the same positive number results in an equivalent
inequality.
2.
For a negative number c:
If a < b , then ac > bc
e.g. c = −3
2<5
2(−3) > 5(−3)
−6 > −15
and
If a > b , then ac < bc
e.g. c = −3
−5 > −12
−5(−3) < −12(−3)
15 < 36
In words this says multiplying both sides of an inequality the same negative
number and reversing the direction of the inequality results in an equivalent
inequality. Similarly, since division is simply multiplication by the
reciprocal, dividing both sides of an inequality by the same negative
number and reversing the direction of the inequality results in an equivalent
inequality.
3.5 Linear Inequalities in One Variable (Page 21 of 24) Example 2
1. Solve 3x − 5 < 7
2. Solve 8 − 2x > 11
3. Solve −3(4 − x) ≥ 14 + x
4. Solve 2[x − (2x + 3)] ≤ 7 − 2(5 − x)
5.
Solve −3[7 − (2 − 4x)] + 12 < 7x − (3x + 8)
3.5 Linear Inequalities in One Variable (Page 22 of 24) Graphs of Solution Sets and Interval Notation
Since the solution set to a linear inequality is generally an infinite set, we
can express the solution set as an inequality, a graph, or as an interval using
what is referred to as interval notation.
In Words
Inequality
Numbers less
than 4
Numbers less
than or equal to 4
Numbers greater
than 4
Numbers greater
than or equal to 4
Numbers
between 0 and 4
Numbers
between 0 and 4,
including 4
Numbers
between 0 and 4,
including 0 and 4
x<4
Graph
0
4
0
4
0
4
0
4
0<x<4
0
4
0<x≤4
0
4
0≤x≤4
0
4
x≤4
x>4
x≥4
Interval
Notation
Example 3
Solve −3(4x − 5) − 1 ≤ 17 − 6x . Express the solution set as an inequality, a
graph, and in interval notation.
3.5 Linear Inequalities in One Variable (Page 23 of 24) Example 4
In February 2004, one Budget office rented pickup trucks for $39.95 per
day plus $0.19 per mile. One U-Haul location charged $19.95 per day plus
$0.49 per mile. For what mileage did Budget offer the lower price?
Example 5
The student-to-teacher ratios for
public and private elementary
schools are given in the table. Let
f (t) and g(t) represent the student
to teacher ratios for public and
private elementary schools,
respectively, at t years since 1990.
1. Verify the regression equations for f and g are f (t) = − 0.22t + 20.16
and g(t) = −0.055t + 16.915 .
2. Symbolically predict the years in which public schools will have the
smaller student-to-teacher ratio.
3.5 Linear Inequalities in One Variable (Page 24 of 24) Example 6
The graphs of functions f and g are given. Answer the following.
a. Find f (− 6) .
y
b. Find x where g(x) = 0 .
5
g
-5
x
5
c. Find the equation for f.
-5
f
y
d. Find the equation for g.
4
2
x
-4
2
-2
e. Find x where f (x) ≤ g(x) . Express
the solution set as an inequality and
in interval notation.
-4
4