Download State whether each sentence is true or false. If false, replace the

Study Guide and Review - Chapter 6
State whether each sentence is true or false. If false, replace the underlined term to make a true
sentence.
1. If a system has at least one solution, it is said to be consistent.
SOLUTION: If a system has at least one solution, it is said to be consistent. So, the statement is true.
2. If a consistent system has exactly two solution(s), it is said to be independent.
SOLUTION: The statement is false. If a consistent system has exactly one solution(s), it is said to be independent.
3. If a consistent system has an infinite number of solutions, it is said to be inconsistent.
SOLUTION: The statement is false. If a consistent system has an infinite number of solutions, it is said to be dependent.
4. If a system has no solution, it is said to be inconsistent.
SOLUTION: If a system has no solution, it is said to be inconsistent. So, the statement is true.
5. Substitution involves substituting an expression from one equation for a variable in the other.
SOLUTION: Substitution involves substituting an expression from one equation for a variable in the other. So, the statement is
true.
6. In some cases, dividing two equations in a system together will eliminate one of the variables. This process is called
elimination.
SOLUTION: The statement is false. In some cases, when adding or subtracting two equations in a system together will eliminate
one of the variables, this process is called elimination.
7. A set of two or more inequalities with the same variables is called a system of equations.
SOLUTION: The statement is false. A set of two or more inequalities with the same variables is called a system of inequalities.
8. When the graphs of the inequalities in a system of inequalities do not intersect, there are no solutions to the system.
SOLUTION: True
Graph each system and determine the number of solutions that it has. If it has one solution, name it.
9. x − y = 1
x +y = 5
SOLUTION: To graph the system, write both equations in slope-intercept form.
Equation 1:
eSolutions Manual - Powered by Cognero
Page 1
The statement is false. A set of two or more inequalities with the same variables is called a system of inequalities.
8. When the graphs of the inequalities in a system of inequalities do not intersect, there are no solutions to the system.
SOLUTION: Study
Guide and Review - Chapter 6
True
Graph each system and determine the number of solutions that it has. If it has one solution, name it.
9. x − y = 1
x +y = 5
SOLUTION: To graph the system, write both equations in slope-intercept form.
Equation 1:
Equation 2:
Graph and locate the solution.
y=x−1
y = −x + 5
The graphs appear to intersect at the point (3, 2). You can check this by substituting 3 for x and 2 for y.
The solution is (3, 2).
12. −3x + y = −3
y =x−3
SOLUTION: To graph the system, write both equations in slope-intercept form.
eSolutions Manual - Powered by Cognero
Equation 1: Page 2
Study
Guide
andisReview
The
solution
(3, 2). - Chapter 6
12. −3x + y = −3
y =x−3
SOLUTION: To graph the system, write both equations in slope-intercept form.
Equation 1: Graph and find the solution.
y = 3x − 3
y=x−3
The graphs appear to intersect at the point (0, −3). You can check this by substituting 0 for x and −3 for y.
The solution is (0, −3).
15. MAGIC NUMBERS Sean is trying to find two numbers with a sum of 14 and a difference of 4. Define two
variables, write a system of equations, and solve by graphing.
SOLUTION: Sample answer: Let x be one number and y be the other number.
x + y = 14
x −y = 4
To graph the system, write both equations in slope-intercept form.
eSolutions Manual - Powered by Cognero
Equation 1:
Page 3
Guide and Review - Chapter 6
Study
The solution is (0, −3).
15. MAGIC NUMBERS Sean is trying to find two numbers with a sum of 14 and a difference of 4. Define two
variables, write a system of equations, and solve by graphing.
SOLUTION: Sample answer: Let x be one number and y be the other number.
x + y = 14
x −y = 4
To graph the system, write both equations in slope-intercept form.
Equation 1:
Equation 2:
Graph the equations and find the solution.
y = −x + 14
y=x−4
The graphs appear to intersect at the point (9, 5). So, the numbers 9 and 5 have a sum of 14 and a difference of 4.
Use substitution to solve each system of equations.
16. x + y = 3
x = 2y
eSolutions Manual - Powered by Cognero
SOLUTION: x +y = 3
Page 4
Guide and Review - Chapter 6
Study
The graphs appear to intersect at the point (9, 5). So, the numbers 9 and 5 have a sum of 14 and a difference of 4.
Use substitution to solve each system of equations.
16. x + y = 3
x = 2y
SOLUTION: x +y = 3
x = 2y
Substitute 2y for x in the first equation.
Use the solution for y and either equation to find x.
x = 2y
x = 2(1)
x =2
The solution is (2, 1).
18. 3x + 2y = 16
x = 3y − 2
SOLUTION: 3x + 2y = 16
x = 3y − 2
Substitute 3y − 2 for x in the first equation.
Use the solution for y and either equation to find x.
The solution is (4, 2).
eSolutions
Manual
= 9 - Powered by Cognero
21. x + 3y
x +y = 1
Page 5
Guide and Review - Chapter 6
Study
The solution is (4, 2).
21. x + 3y = 9
x +y = 1
SOLUTION: x + 3y = 9
x +y = 1
First, solve the second equation for y to get y = −x +1. Then substitute −x + 1 for y in the first equation.
Use the solution for x and either equation to find y.
The solution is (−3, 4).
22. GEOMETRY The perimeter of a rectangle is 48 inches. The length is 6 inches greater than the width. Define the
variables, and write equations to represent this situation. Solve the system by using substitution.
SOLUTION: Sample answer: Let w be the width and be the length.
2 + 2w = 48
=w+6
Substitute w + 6 for in the first equation.
Use the solution for w and either equation to find .
=w+6
=9+6
= 15
The solution is (9, 15).
Use Manual
elimination
each
eSolutions
- Poweredto
bysolve
Cognero
24. −3x + 4y = 21
3x + 3y = 14
system of equations.
Page 6
=w+6
=9+6
= 15
Study
Guide and Review - Chapter 6
The solution is (9, 15).
Use elimination to solve each system of equations.
24. −3x + 4y = 21
3x + 3y = 14
SOLUTION: Because −3x and 3x have opposite coefficients, add the equations.
Now, substitute 5 for y in either equation to find x.
The solution is
.
27. 6x + y = 9
−6x + 3y = 15
SOLUTION: Because 6x and −6x have opposite coefficients, add the equations.
Now, substitute 6 for y in either equation to find x.
The solution is
.
eSolutions Manual - Powered by Cognero
30. 3x + 2y = 8
x + 2y = 2
Page 7
The
solution
Study
Guide
andisReview - .Chapter 6
27. 6x + y = 9
−6x + 3y = 15
SOLUTION: Because 6x and −6x have opposite coefficients, add the equations.
Now, substitute 6 for y in either equation to find x.
The solution is
.
30. 3x + 2y = 8
x + 2y = 2
SOLUTION: Because 2y and 2y have same coefficients, multiply equation 2 by –1 so the terms are additive inverses. Then add
the equations.
Now, substitute 3 for x in either equation to find y.
eSolutions Manual - Powered by Cognero
The solution is
.
Page 8
Study
Guide
andisReview .- Chapter 6
The
solution
30. 3x + 2y = 8
x + 2y = 2
SOLUTION: Because 2y and 2y have same coefficients, multiply equation 2 by –1 so the terms are additive inverses. Then add
the equations.
Now, substitute 3 for x in either equation to find y.
The solution is
.
Use elimination to solve each system of equations.
33. x − y = −2
2x + 4y = 38
SOLUTION: Notice that if you multiply the first equation by 4, the coefficients of the y-terms are additive inverses.
Now, substitute 5 for x in either equation to find y.
The Manual
solution
is (5, 7).
eSolutions
- Powered
by Cognero
36. 8x − 3y = −35
Page 9
Study
Guide
andisReview - .Chapter 6
The
solution
Use elimination to solve each system of equations.
33. x − y = −2
2x + 4y = 38
SOLUTION: Notice that if you multiply the first equation by 4, the coefficients of the y-terms are additive inverses.
Now, substitute 5 for x in either equation to find y.
The solution is (5, 7).
36. 8x − 3y = −35
3x + 4y = 33
SOLUTION: Notice that if you multiply the first equation by 4 and the second equation by 3, the coefficients of the y-terms are
additive inverses.
Now, substitute −1 for x in either equation to find y.
The solution is (−1, 9).
39. 8x − 5y = 18
6x + 6y = −6
SOLUTION: Notice that if you multiply the first equation by 6 and the second equation by 5, the coefficients of the y-terms are
eSolutions
Manual
- Powered by Cognero
Page 10
additive
inverses.
Study
Guide and Review - Chapter 6
The solution is (−1, 9).
39. 8x − 5y = 18
6x + 6y = −6
SOLUTION: Notice that if you multiply the first equation by 6 and the second equation by 5, the coefficients of the y-terms are
additive inverses.
Now, substitute 1 for x in either equation to find y.
The solution is (1, −2).
Determine the best method to solve each system of equations. Then solve the system.
42. y = −x
y = 2x
SOLUTION: Because both equations are solved for one of the variables, substitution is the best method.
Substitute 2x for y in the first equation.
Substitute 0 for x in either equation to find y.
The solution is (0, 0).
45. 3x + 2y = −4
5x + 2y = −8
SOLUTION: Because 2y and 2y have the same coefficient, elimination using addition is the best method. Multiply equation 2 by –
eSolutions
Page 11
1. Manual - Powered by Cognero
Study
Guide
andisReview
The
solution
(0, 0). - Chapter 6
45. 3x + 2y = −4
5x + 2y = −8
SOLUTION: Because 2y and 2y have the same coefficient, elimination using addition is the best method. Multiply equation 2 by –
1. Now, substitute −2 for x in either equation to find y.
The solution is (−2, 1).
48. 11x − 6y = 3
5x − 8y = −25
SOLUTION: Because none of the coefficients are 1 or −1, elimination using multiplication is the best method.
Notice that if you multiply the first equation by 4 and the second equation by −3, the coefficients of the y-terms are
additive inverses.
Now, substitute 3 for x in either equation to find y.
eSolutions Manual - Powered by Cognero
The solution is (3, 5).
Page 12
Study
Guide and Review - Chapter 6
The solution is (−2, 1).
48. 11x − 6y = 3
5x − 8y = −25
SOLUTION: Because none of the coefficients are 1 or −1, elimination using multiplication is the best method.
Notice that if you multiply the first equation by 4 and the second equation by −3, the coefficients of the y-terms are
additive inverses.
Now, substitute 3 for x in either equation to find y.
The solution is (3, 5).
Solve each system of inequalities by graphing.
51. x > 3
y <x+2
SOLUTION: Graph each inequality. The graph of x > 3 is dashed and is not included in the graph of the solution. The graph of y < x + 2 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of x > 3 and y < x + 2. Overlay
Page 13
the graphs and locate the green region. This is the intersection.
eSolutions Manual - Powered by Cognero
Study
Guide and Review - Chapter 6
The solution is (3, 5).
Solve each system of inequalities by graphing.
51. x > 3
y <x+2
SOLUTION: Graph each inequality. The graph of x > 3 is dashed and is not included in the graph of the solution. The graph of y < x + 2 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of x > 3 and y < x + 2. Overlay
the graphs and locate the green region. This is the intersection.
The solution region is shaded in the graph below.
54. y ≤ −x − 3
y ≥ 3x − 2
eSolutions Manual - Powered by Cognero
SOLUTION: Graph each inequality.
Page 14
Study Guide and Review - Chapter 6
54. y ≤ −x − 3
y ≥ 3x − 2
SOLUTION: Graph each inequality.
The graph of y ≤ −x − 3 is solid and is included in the graph of the solution. The graph of y ≥ 3x − 2 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ≤ −x − 3 and y ≥ 3x − 2.
Overlay the graphs and locate the green region. This is the intersection.
The solution region is shaded in the graph below.
eSolutions Manual - Powered by Cognero
Page 15