Download Chapter 13 — Systems of Equations and Inequalities

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Graphing Systems of
Equations (Pages 550–553)
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A set of equations with the same variables forms a system of equations. A
solution to a system of two or more equations is an ordered pair of numbers that
satisfies all of the equations. One way to solve a system of equations is to
carefully graph the equations on the same coordinate plane. The coordinates of
the point at which the graphs intersect are the solution to the system.
EXAMPLE
Graph the system of equations to find the solution.
y 2x 3 and y x 1
y
The graphs appear to intersect at the point with coordinates (2, 1).
Check this estimate by replacing x with 2 and y with 1 in each equation.
Check: y 2x 3
yx1
1 2(2) 3
121
11✓
11✓
The solution is (2, 1).
(2, 1)
x
O
y=x–1
y = 2x – 3
Try These Together
Solve each system of equations by graphing.
1. y x 2
y 2x 1
2. y 2x 2
yx1
3. y 2x 1
y 3x 3
HINT: Be sure to check your solution by substituting the x- and y-values back into the two equations.
PRACTICE
Solve each system of equations by graphing.
4. y 9 x
5. 2x y 5
yx1
3x 3y 9
7. y 3
4x y 1
3
10. y x1
2
y 2x 6
B
4.
1
y
x1
2
xy4
11. 2x 3y 6
12. 2x y 2
3x y 2
xy3
C
B
8.
9. y x 2
C
B
A
7.
8. x y 2
C
A
5.
6.
1
y4
x
3
B
A
13. Standardized Test Practice Solve the system of equations.
xy2
x y 10
A (3, 7)
B (7, 3)
C (4, 6)
D (6, 4)
Answers: 1–12. See Answer Key for graphs. 1. (3, 5) 2. (3, 4) 3. (2, 3) 4. (4, 5) 5. (8, 11) 6. (6, 2) 7. (1, 3) 8. (3, 1)
9. (2, 0) 10. (2, 2) 11. (0, 2) 12. (1, 4) 13. D
3.
6. y 8 x
© Glencoe/McGraw-Hill
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CA Parent and Student Study Guide
Algebra: Concepts and Applications
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Solutions of Systems of
Equations (Pages 554–559)
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Graphs of systems of equations may be intersecting lines, parallel lines, or
the same line. Systems of equations can be described by the number of
solutions they have. A consistent system has at least 1 solution. An
inconsistent system has no solution.
y
y
(–1, 5)
y
x+y=2
y=x+2
x – y = –6
Graph
2x + y = 3
x
O
x
O
y=x–1
x
O
3x + 3y = 6
Description
of Graph
intersecting lines
same line
parallel lines
Number of
Solutions
1
infinitely many
0
consistent and
independent
consistent and
dependent
inconsistent
Type of
System
PRACTICE
State whether each system is consistent and independent, consistent and
dependent, or inconsistent.
1.
2.
y
3.
y
y
x
O
y = 2x – 1
y = –4x
x
O
y = –4x – 6
O
y = 3x + 2
x
6x + 2y = 4
4x – 2y = 2
Determine whether each system of equations has one solution, no solution, or
infinitely many solutions by graphing. If the system has one solution, name it.
4. y x 4
1
x
2
B
C
B
8.
2
7. y 2x 4
3x y 5
C
B
A
7.
x
1
y
3
C
A
5.
6.
xy5
B
A
8. Standardized Test Practice What is the solution of the system y x 3 and
y 2x 9?
A Infinite solutions
B no solution
C (2, 5)
3. consistent and dependent 4–7. See Answer Key for graphs. 4. (4, 0)
4.
6. y 3x 6
© Glencoe/McGraw-Hill
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D (5, 2)
Answers: 1. inconsistent 2. consistent and independent
5. no solution 6. infinitely many 7. (1, 2) 8. C
3.
y2
5. y x 6
CA Parent and Student Study Guide
Algebra: Concepts and Applications
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Substitution (Pages 560–565)
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To solve a system of equations without graphing, you can use the substitution
method shown in the example below. In general, if you solve a system of
equations and the result is a true statement, such as 5 5, the system
has infinitely many solutions; if the result is a false statement, such as 5 7,
the system has no solution.
EXAMPLE
Use substitution to solve the system of equations x y 1 and 2x y 1.
Step 1: Solve one of the equations for x or y.
xy1
Solve the first equation for x since the
x 1 y coefficient of x is 1.
Step 3: Solve this equation.
2 2y y 1
y 3 or y 3
Step 2: Substitute this value into the other equation.
2x y 1 Use the second equation.
2(1 y) y 1 Substitute 1 y for x.
2 2y y 1 Distributive Property
Step 4: Find the value of the other variable using
substitution into either equation.
xy1
Use the first equation.
x31
Substitute 3 for y.
x 2 Solve for x.
The solution to the system is (2, 3).
Check: Substitute 2 for x and 3 for y in each of the
original equations and check for true statements.
Try These Together
Use substitution to solve each system of equations.
1. 3x y 19
x 2y 10
2. 2x y 7
8x y 3
3. y 2x 4
y 2x 2
4. y 5x 3
y 3x 3
HINT: If possible, choose to first solve an equation for a variable that has a coefficient of 1.
PRACTICE
Use substitution to solve each system of equations.
5. 5x 4 y
y 3x 7
9. 2x y 4
x y 9
B
12. 5x y 98
2x 3y 5
C
13. Standardized Test Practice All CDs in the budget bin are priced the same.
Packs of AA batteries are on sale. Keisha’s total bill (before tax) for 3 CDs
and 1 pack of AA batteries was $39. Eduardo’s total for 2 CDs and
3 packs of batteries was $33. What was the price of a single CD?
A $3
B $10
C $12
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© Glencoe/McGraw-Hill
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B
A
93
3
B
8.
11. 3x y 28
x 3y 12
C
B
A
7.
10. 5x 2y 6
2x 3y 9
C
A
5.
6.
8. 3y 4x 2
8x 6y 4
D $13
3
4.
7. 6x y 0
3x 4y 18
Answers: 1. (4, 7) 2. (1, 5) 3. no solution 4. , 5. , 6. 5, 7. , 4 8. infinitely many 9. (13, 22)
4
4
2 2 3
3
10. (0, 3) 11. (12, 8) 12. (23, 17) 13. C
3.
6. 3y x 1
2x 6 3y
CA Parent and Student Study Guide
Algebra: Concepts and Applications
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Elimination Using Addition
and Subtraction (Pages 566–571)
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In systems of equations where the coefficients of terms containing the same
variable are opposites, the elimination method can be applied by adding the
equations. If the coefficients of those terms are the same, the elimination
method can be applied by subtracting the equations.
EXAMPLES
Solve each system of equations using elimination.
A x 2y 13 and 3x 2y 15
B 3x 4y 5 and 3x y 5
Add the two equations, since the coefficients of
the y-terms, 2 and 2, are opposites.
Subtract the two equations, since the coefficients
of the x-terms are the same.
x 2y 13
() 3x 2y 15
4x 0 28
x7
Solve for x.
3x 4y 5
() 3x y 5
0 5y 10
y2
Solve for y.
x 2y 13
Use the first equation.
7 2y 13
Substitute 7 for x.
2y 6 ⇒ y 3
3x y 5
Use the second equation.
3x 2 5
Substitute 2 for y.
3x 3 ⇒ x 1
The solution of the system is (7, 3).
The solution of the system is (1, 2).
Try These Together
Use elimination to solve each system of equations
1. x y 3
3x y 1
2. 3x 4y 2
2x 4y 8
3. 2x 4y 8
y 7 2x
PRACTICE
Use elimination to solve each system of equations.
4. x 2y 3
x y 6
5. x y 2
xy8
6. 2y 3x 12
2y 6x 5
8. x 4y 16
2x 4y 18
9. 2x 4y 12
3x 4y 8
10. 8x y 4
8x 4y 8
7. 2x y 5
2x 3y 25
11. 2x 5y 7
2x 3y 1
12. Number Theory The sum of two numbers is 22. The greater number is
two less than twice the other number. Write a system of equations to
represent the problem. What are the numbers?
B
C
C
8.
C
B
A
13. Standardized Test Practice Solve the system. 2y 5x 1
3y 5x 14
A (3, 1)
B (1, 3)
7
A
7.
C (1, 3)
© Glencoe/McGraw-Hill
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B
B
6.
D (3, 1)
7
A
5.
Answers: 1. (1, 2) 2. (6, 5) 3. (2, 3) 4. (3, 3) 5. (3, 5) 6. , 7. (5, 5) 8. 2, 9. (4, 1) 10. (1, 4)
3 2 2
4.
11. (1, 1) 12. x y 22 and x 2y 2; 14, 8 13. B
3.
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Algebra: Concepts and Applications
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Elimination Using
Multiplication (Pages 572–577)
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An extension of the elimination method is to multiply one or both of the
equations in a system by some number so that adding or subtracting
eliminates a variable.
EXAMPLES
Solve each system of equations using elimination.
A x y 5 and 3x 2y 15
B 2x 9y 43 and 5x 2y 15
Multiply the first equation by 2 so that the
coefficients of the y-terms in the system will be
opposites. Then, add the equations and solve for x.
2(x y) 2(5)
3x 2y 15
➞
Multiply the first equation by 5 and the second
equation by 2 so that the coefficients of the x-terms
in the system will be opposites. Then, add the
equations and solve for y.
2x 2y 10
() 3x 2y 15
5x 0 25
x5
5(2x 9y) 5(43)
2(5x 2y) 2(15)
xy5
Use the first equation.
5y5
Substitute 5 for x.
y 0 ⇒ y 0
➞
2x 9y 43
2x 45 43
2x 2 ⇒ x 1
The solution to this system is (5, 0).
10x 45y 215
() 10x 4y 30
0 49y 245
y5
Use the first equation.
Substitute 5 for y.
The solution to the system is (1, 5).
Try These Together
Use elimination to solve each system of equations.
1. 2x y 4
3x 2y 6
2. 5x 2y 5
xy2
3. 4x 7y 6
6x 5y 20
4. x y 4
4x 2y 24
PRACTICE
Use elimination to solve each system of equations.
5. 3x 4y 48
4x 3y 43
9. 2x 3y 0
3x y 7
6. 3x 8y 13
2x 5y 7
7. y 4x 11
3x 2y 7
1
10. 2x y 1
3
8. 2x 2y 16
3x y 4
11. 4x 2y 4
2x 3y 4
1
x
y 8
4
12. Solve using elimination: 2x y 4 and x 2y 1.
B
C
C
B
C
8.
B
A
13. Standardized Test Practice By which number could you multiply the first
equation of the following system to solve the system by elimination?
4x 11y 32 and 12x 10y 55
A 3 or 3
B 10 or 10
C 11 or 11
D 12 or 12
7
A
7.
1
B
6.
5
A
5.
Answers: 1. (2, 0) 2. (3, 5) 3. (5, 2) 4. (8, 4) 5. (4, 9) 6. (9, 5) 7. (3, 1) 8. (3, 5) 9. (3, 2) 10. , 18
2
4.
11. , 12. (3, 2) 13. A
4
2
3.
© Glencoe/McGraw-Hill
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CA Parent and Student Study Guide
Algebra: Concepts and Applications
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Solving Quadratic-Linear Systems
of Equations (Pages 580–585)
A quadratic-linear system of equations can have 0, 1, or 2 solutions. You
can use graphing and substitution to solve quadratic-linear systems.
y
y
y
(–2, 2)
x
O
O
x
O
(1, –2)
no solution
one solution
(1, –1) x
two solutions
EXAMPLE
Solve. y x2 5
yx3
Substitute the values of x into one of the original
equations to find values of y. Choose the easier equation
to solve.
Substitute x 3 for y in the first equation.
x 3 x2 5
x x2 2
0 x2 x 2
0 (x 1)(x 2)
0x1
1 x
or
yx3
y 1 3
y 4
Add 3 to each side.
Subtract x from each side.
Factor.
yx3
y23
y1
y
The solutions are (1, 4)
and (2, 1). The graph
shows the solutions are
correct.
0x2
2x
x
(2, –1)
O
(–1, –4)
y = x2 – 5
y=x–3
PRACTICE
Solve each system of equations by graphing.
1. y x2 3
y x
2. y x2 4
1 2
4. y x 1
2
3. y 5
y 2x2 5
yx2
y x 1
Use substitution to solve each system of equations.
5. x 2
y x2 1
B
4.
C
B
C
B
A
7.
8.
8. y x 4
y x2 8
C
A
5.
6.
7. y 2x 4
y x2 x 2
B
A
9. Standardized Test Practice What is the solution of the system y 3x 3
and y x2 1?
A (1, 0), (4, 15)
B (0, 1), (15, 4)
C no solution
D (1, 0)
Answers: 1–4. See Answer Key for graphs. 1. no solution 2. (2, 0), (3, 5) 3. (0, 5) 4. (2, 3), (0, 1) 5. (2, 3) 6. no solution
7. (2, 0), (3, 10) 8. (3, 1), (4, 8) 9. A
3.
6. y x2 5
yx1
© Glencoe/McGraw-Hill
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CA Parent and Student Study Guide
Algebra: Concepts and Applications
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Graphing Systems of
Inequalities (Pages 586–590)
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You can solve systems of inequalities by graphing. Recall that the graph of
an inequality is a half-plane. The intersection of the two half-planes graphed
in a system of inequalities represents the solution to the system.
EXAMPLE
Graph the system of inequalities to find the solution.
x y 3 and y 3 x
y
y = –x + 3
Begin by solving each inequality for y. Then, graph each inequality.
xy3
y x 3
and
y3x
yx3
x
O
y=x–3
The solution to the system includes the ordered pairs in the intersection
of the graphs of each inequality. This region is shaded dark gray.
Notice that the boundary lines y x 3 and y x 3 are included
in the solution, since the inequalities contained and symbols.
Try These Together
Solve each system of inequalities by graphing. If the system does not
have a solution, write no solution.
1. x 3
y5
2. x 4
y 1
3. y 3 x
yx3
4. 2y x 6
3x y 4
HINT: Remember to graph inequalities with or with dashed lines because these lines are not
included in the solution.
PRACTICE
Solve each system of inequalities by graphing. If the system does not have a
solution, write no solution.
5. x 1
y 4
6. 2x y 4
3x y 6
7. y 2 x
2y 2 2x
8. x 4 y
y
2
9. Money Carlita has some nickels and dimes in her pocket. She has at
most 10 coins and they are worth at most 90 cents total. What
combination of coins could she have?
B
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
10. Standardized Test Practice A dieter limits a snack to 90 Calories. Which is a
possible snack combination of 20-Calorie apricots and 3-Calorie celery stalks?
A 4 apricots
3 celery stalks
B 3 apricots
10 celery stalks
10. D
4.
© Glencoe/McGraw-Hill
C 2 apricots
8 celery stalks
D all of these
Answers: 1–8. See Answer Key. 9. Sample answer: 5 nickels, 5 dimes
3.
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Algebra: Concepts and Applications
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Chapter 13 Review
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Treasure Hunt
Imagine that you and your parent are on a treasure hunt. The treasure hunt
is taking place on a giant coordinate grid that is laid out on the floor of your
school gym. You are competing with other parents and students for a grand
prize. However, every parent/student team is looking for different treasures.
The treasures for which you are searching are numbered stickers on the floor
of this giant coordinate grid. Specifically, you are given a list of four items
and a starting point. To locate the treasures, you must plot the intersection
of the two graphs listed for each treasure.
2
Your starting point is the intersection of the graphs of y x 5 and
3
2y x 10. Find the coordinates of your starting point by graphing the two
equations. Determine the coordinates for the location of each treasure by
graphing each pair of equations given in the figure below. Then determine
which treasure is closest to your starting point. The winner of the treasure
hunt is the first parent/student team who turns in the treasure sticker that
was closest to their starting point.
1
Gold: y 2x 8 and y x3
2
Silver: 3x 2y 6 and x 2y 10
1
1
Diamonds: x y 1 and y x7
4
4
Jewels: x 2y – 1 and y x 1
N
Answers are located on page 121.
© Glencoe/McGraw-Hill
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CA Parent and Student Study Guide
Algebra: Concepts and Applications