Transcript Chapter 4-3

Chapter 4
Systems of
Equations and
Inequalities
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 4-1
1
Chapter Sections
4.1 – Solving Systems of Linear Equations in Two
Variables
4.2 – Solving Systems of Linear Equations in
Three Variables
4.3 – Systems of Linear Equations: Applications
and Problem Solving
4.4 – Solving Systems of Equations Using
Matrices
4.5 – Solving Systems of Equations Using
Determinants and Cramer’s Rule
4.6 – Solving Systems of Linear Inequalities
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 4-2
2
§ 4.3
Systems of
Linear Equations:
Applications and
Problem Solving
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 4-3
3
Use Systems of Equations to Solve Applications
Example
The combined land area of North Carolina and
South Carolina is 78,820 square miles. The
difference between the two state’s land areas is
18, 602 square miles. If North Carolina has the
larger land area, find the land area of each state.
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Chapter 4-4
4
Use Systems of Equations to Solve Applications
Solution
Understand We need to determine the land
area of North Carolina and the land area of
South Carolina. We will use two variables and
therefore will need to determine two equations.
Translate
Let N= the land area of North Carolina
Let S = the land area of South Carolina
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 4-5
5
Use Systems of Equations to Solve Applications
Since the total area of the two states is 78, 820
square miles, the first equation is
N + S = 78,820
Since North Carolina has a larger land area and
since the difference in the land area is 18,602
square miles, the second equation is
N – S = 18,602
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Chapter 4-6
6
Use Systems of Equations to Solve Applications
The system of two equations is
N + S = 78,820
N – S = 18,602
Carry Out We will use the addition method to
solve this system of equations.
N + S = 78,820
N – S = 18,602
2N = 97,422
N = 48,711
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Chapter 4-7
7
Use Systems of Equations to Solve Applications
Thus, N = 48,711. To determine the value for S,
substitute 48,711 into the equation.
N + S = 78,820
48,711 + S = 78,820
S = 30,109
Answer The land area of North Carolina is
48,711 square miles and the land area of South
Carolina is 30,109 square miles.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 4-8
8
Using Tables
Example Chung Song, a chemist with Johnson
and Johnson, wishes to create a new household
cleaner containing 30% trisodium phosphate
(TSP). Chung needs to mix a 16% TSP solution
with a 72% TSP solution to get 6 liters of 30% TSP
solution. How many liters of the 16% solution
and of the 72% solution will he need to mix?
Use a table to organize your information.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 4-9
9
Using Tables
Let x = number of liters of the 16% solution
Let y = number of liters of the 72% solution
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 4-10
10
Using Tables
Solution
16%
solution
72%
solution
Mixture
Strength of Number of
Solution
Liters
Amount of
TSP
0.16
x
0.16x
0.72
y
0.72y
0.30
6
0.30(6)
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Chapter 4-11
11
Using Tables
Since the sum of the volumes of the 16% solution
and 72% solution is 6 liters, our first equation is
x+y=6
The second equation comes form the fact that the
solutions are mixed.
Amount of TSP
in 16% solution
+
Amount of TSP
in 72% solution
=
Amount of TSP
in mixture
Therefore, the systems of equation is
x+y=6
0.16x + 0.72y =- 0.30(6)
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Chapter 4-12
12
Using Tables
Carry Out Solving x + 6 = 6 for y, we get y = -x + 6.
Substituting –x + 6 for y in the second equation gives us
0.16 x  0.72 y  0.30(6)
0.16 x  0.72( x  6)  0.30(6)
0.16 x  0.72 x  4.32  1.8
 0.56 x  2.52
 2.52
x
 4.5
 0.56
Therefore, Chung must use 4.5 liters of the 16%
solution. Since the two solutions must total 6 liters,
he must use 6 – 4.5 or 1.5 liters of the 72% solution
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Chapter 4-13
13
Three Variables
Tiny Tots Toys must borrow $25,000 to pay for an
expansion. It is not able to obtain a loan for the total
amount from a single bank, so it takes out loans from three
different banks. It borrows some of the money at a bank
that charges 8% interest. At the second bank, it borrows
$2,000 more than one-half the amount borrowed form the
first bank. The interest rate at the second bank is 10%.
The balance of the $25,000 is borrowed from a third bank,
where Tiny Tots pays 9% interest. The total annual
interest Tiny Tots Toys pays for the three loans is $2220.
How much does it borrow at each rate?
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Chapter 4-14
14
Three Variables
Let x = amount borrowed at first bank
Let y = amount borrowed at second bank
Let z = amount borrowed at third bank
Since the total amount borrowed is $25,000 we
know that
x + y + z = 25,000
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Chapter 4-15
15
Three Variables
At the second bank, Tiny Tots borrows $2000 more
than one-half the amount borrowed from the first
bank. Therefore, our second equation is
1
y  x  2000
2
Our last equation comes from the fact that the total
annual interest charged by the three banks is $2220.
The interest at each bank is found by multiplying
the interest rate by the amount borrowed.
0.08 x  0.10 y  0.09 z  2220
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Chapter 4-16
16
Three Variables
Thus, our system of equations is
x  y  z  25,000
1
y  x  2000
2
0.08 x  0.10 y  0.09 z  2220
Both sides of equation 2 can be multiplied by 2 to
remove fractions
1

2( y )  2 x  2000 
2

2 y  x  4000
 x  2 y  4000
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Chapter 4-17
17
Three Variables
The decimals in equation 3 can be removed by
multiplying both sides of the equation by 100. This
gives
8x +10y + 9z = 222,000
Our simplified system of equations is therefore
x  y  z  25,000
 x  2 y  4000
8 x  10 y  9 z  222,000
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Chapter 4-18
18
Three Variables
Carry Out There are various ways of solving this system.
Let’s use equation 1 and 3 to eliminate the variable z.
-9x - 9y - 9z = -225,000
8x + 10y + 9z = 222,000
-x + y
= -3,000
Now we use equations 2 and 4 to eliminate the variable x
and solve for y.
x - 2y = -4,000
-x + y = -3,000
-y = -7,000
y=
7,000
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Chapter 4-19
19
Three Variables
Now that we know the value of y we can solve for x.
 x  2 y  4000
 x  2(7000)  4000
 x  14,000  4,000
 x  10,000
x  10,000
Now we solve for z.
x  y  z  25,000
Answer Tiny Tots
10,000  7000  z  25,000
17,000  z  25,000
z  8000
Toys borrows $10,000 at
8%, $7000 at 10%, and
$8000 at 9% interest.
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Chapter 4-20
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