Download Use the graph to determine whether each system is

Mid-Chapter Quiz
Use the graph to determine whether each
system is consistent or inconsistent and if it is
independent or dependent.
SOLUTION: The lines y = −2x + 3 and y = −2x − 3 never intersect
which means this system has no solution. So, the
system is inconsistent.
Graph each system and determine the number
of solutions that it has. If it has one solution,
name it.
3. y = 2x − 3
y =x+4
SOLUTION: y = 2x − 3
y=x+4
1. y = 2x − 1
y = −2x + 3
SOLUTION: The lines y = 2x − 1 and y = −2x + 3 intersect at
exactly one point which means this system has
exactly one solution. So, the system is consistent and
independent.
The graphs appear to intersect at the point (7, 11).
You can check this by substituting 7 for x and 11 for
y.
2. y = −2x + 3
y = −2x − 3
SOLUTION: The lines y = −2x + 3 and y = −2x − 3 never intersect
which means this system has no solution. So, the
system is inconsistent.
Graph each system and determine the number
of solutions that it has. If it has one solution,
name it.
3. y = 2x − 3
y =x+4
SOLUTION: y = 2x − 3
y=x+4
The graphs appear to intersect at the point (7, 11).
You can check this by substituting 7 for x and 11 for
y.
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The solution is (7, 11).
4. x + y = 6
x −y = 4
SOLUTION: To graph the system, write both equations in slopeintercept form.
y = –x + 6
y=x–4
The graphs appear to intersect at the point (5, 1).
You can check this by substituting 5 for x and 1 for
y.
Page 1
Mid-Chapter
Quiz
The solution
is (7, 11).
4. x + y = 6
x −y = 4
SOLUTION: To graph the system, write both equations in slopeintercept form.
y = –x + 6
y=x–4
There are an infinite number of solutions.
6. x − 4y = −6
y = −1
SOLUTION: To graph the system, write both equations in slopeintercept form.
Graph
and y = –1.
The graphs appear to intersect at the point (5, 1).
You can check this by substituting 5 for x and 1 for
y.
The solution is (5, 1).
The graphs appear to intersect at the point (–10, –1).
You can check this by substituting –10 for x and –1
for y.
5. x + y = 8
3x + 3y = 24
SOLUTION: To graph the system, write both equations in slopeintercept form.
y = –x + 8
y = –x + 8
When written in slope-intercept form, you can see
that the equations represent the same line.
The solution is (–10, –1).
7. 3x + 2y = 12
3x + 2y = 6
SOLUTION: To graph the system, write both equations in slopeintercept form.
Equation 1: There are an infinite number of solutions.
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6. x − 4y
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y = −1
SOLUTION: Equation 2: Page 2
Mid-Chapter Quiz
The solution is (–10, –1).
The lines are parallel. So, there is no solution.
7. 3x + 2y = 12
3x + 2y = 6
8. 2x + y = −4
5x + 3y = −6
SOLUTION: To graph the system, write both equations in slopeintercept form.
Equation 1: SOLUTION: To graph the system, write both equations in slopeintercept form.
Equation 1:
Equation 2:
Equation 2: Graph y = –2x − 4 and
.
Graph
and
.
The graphs appear to intersect at the point (–6, 8).
You can check this by substituting –6 for x and 8 for
y.
The lines are parallel. So, there is no solution.
8. 2x + y = −4
5x + 3y = −6
SOLUTION: To graph the system, write both equations in slopeintercept form.
Equation 1:
Equation 2:
The solution is (–6, 8).
Use substitution to solve each system of
equations.
9. y = x + 4
2x + y = 16
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Graph y = –2x − 4 and
.
SOLUTION: y=x+4
2x + y = 16
Page 3
Mid-Chapter Quiz
The solution is (–6, 8).
Use substitution to solve each system of
equations.
9. y = x + 4
2x + y = 16
SOLUTION: y=x+4
2x + y = 16
Substitute x + 4 for y in the second equation.
Use the solution for x and either equation to find the
value for y.
The solution is (4, 8).
10. y = −2x − 3
x +y = 9
SOLUTION: y = −2x − 3
x +y = 9
Substitute −2x − 3 for y in the second equation.
The solution is (4, 8).
10. y = −2x − 3
x +y = 9
SOLUTION: y = −2x − 3
x +y = 9
Substitute −2x − 3 for y in the second equation.
Use the solution for x and either equation to find the
value for y.
The solution is (–12, 21).
11. x + y = 6
x −y = 8
SOLUTION: x +y = 6
x −y = 8
Solve the first equation for x.
Substitute 6 – y for x in the second equation.
Use the solution for x and either equation to find the
value for y.
Use the solution for y and either equation to find the
value for x.
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solution
is (–12,by21).
11. x + y = 6
Page 4
Mid-Chapter
Quiz
The solution is (–12, 21).
11. x + y = 6
x −y = 8
SOLUTION: x +y = 6
x −y = 8
Solve the first equation for x.
Substitute 6 – y for x in the second equation.
The solution is (7, –1).
12. y = −4x
6x − y = 30
SOLUTION: y = −4x
6x − y = 30
Substitute −4x for y in the second equation.
Use the solution for x and either equation to find the
value for y.
Use the solution for y and either equation to find the
value for x.
The solution is (3, –12).
13. FOOD The cost of two meals at a restaurant is
shown in the table.
The solution is (7, –1).
12. y = −4x
6x − y = 30
SOLUTION: y = −4x
6x − y = 30
Substitute −4x for y in the second equation.
a. Define variables to represent the cost of a taco
and the cost of a burrito.
b. Write a system of equations to find the cost of a
single taco and a single burrito.
c. Solve the systems of equations, and explain what
the solution means.
Use the solution for x and either equation to find the
value for y.
d. How much would a customer pay for 2 tacos and
2 burritos?
SOLUTION: a. Let t = the cost of a taco and b = the cost of a
burrito.
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The solution is (3, –12).
13. FOOD The cost of two meals at a restaurant is
b. The cost of a meal with 3 tacos and 2 burritos is
$7.40. So, 3t + 2b = 7.4. The cost of a meal with 4
tacos and 1 burrito is $6.45. So, 4t + b = 6.45. Page 5
c. 3t + 2b = 7.4
a. Let t = the cost of a taco and b = the cost of a
burrito.
Mid-Chapter Quiz
b. The cost of a meal with 3 tacos and 2 burritos is
$7.40. So, 3t + 2b = 7.4. The cost of a meal with 4
tacos and 1 burrito is $6.45. So, 4t + b = 6.45.
c. 3t + 2b = 7.4
4t + b = 6.45
Solve the second equation for b.
a. Define variables to represent the cost of an adult
ticket and the cost of a child ticket.
b. Write a system of equations to find the cost of an
adult and child admission.
c. Solve the system of equations, and explain what
the solution means.
Substitute –4t + 6.45 for b in the first equation.
d. How much will a group of 3 adults and 5 children be charged for admission?
SOLUTION: a. Let a = cost of an adult ticket and c = the cost of
a child ticket.
b. The cost of a group with 4 adults and 2 children is
$184. So, 4a + 2c = 184. The cost of a group with 4
adults and 3 children is $200. So, 4a + 3c = 200.
Use the solution for t in either equation to find the
value of b.
Notice that the coefficients of the a terms are the
same, so subtract the equations.
The cost of a single taco is $1.10 and the cost of a
single burrito is $2.05.
Use the solution for c in either equation to find the
value of a.
d. Substitute these values into the equation to find
how much a customer pays for 2 tacos and 2
burritos.
A customer would pay $6.30 for 2 tacos and 2
burritos.
14. AMUSEMENT PARKS The cost of two groups
going to an amusement park is shown in the table.
The cost of an adult’s ticket is $38, and the cost of a
child’s ticket is $16.
d. Substitute these values into the equation to find
the total cost of admission.
A group of 3 adults and 5 children visiting the
amusement park will be charged $194 for admission.
a. Define variables to represent the cost of an adult
ticket and the cost of a child ticket.
b. Write a system of equations to find the cost of an
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adult and child admission.
c. Solve the system of equations, and explain what
15. MULTIPLE CHOICE Angelina spends $16 for
12 pieces of candy to take to a meeting. Each
chocolate bar costs $2, and each lollipop costs $1.
Determine how many of each she bought.
A 6 chocolate bars, 6 lollipops
B 4 chocolate bars, 8 lollipops
Page 6
the total cost of admission.
A group ofQuiz
3 adults and 5 children visiting the
Mid-Chapter
amusement park will be charged $194 for admission.
15. MULTIPLE CHOICE Angelina spends $16 for
12 pieces of candy to take to a meeting. Each
chocolate bar costs $2, and each lollipop costs $1.
Determine how many of each she bought.
Angelina can buy 4 chocolate bars and 8 lollipops. So, B is the correct choice.
Use elimination to solve each system of
equations.
16. x + y = 9
x − y = −3
SOLUTION: Because y and −y have opposite coefficients, add the
equations.
A 6 chocolate bars, 6 lollipops
B 4 chocolate bars, 8 lollipops
C 7 chocolate bars, 5 lollipops
D 3 chocolate bars, 9 lollipops
SOLUTION: Let c = the number of chocolate bars and = the
number of lollipops. Angelina buys 12 pieces of
candy. So, c + = 12. She spends $16. So, 2c + 1
= 16. Since both equations contain , use elimination
by subtraction.
Now, substitute 3 for x in either equation to find the
value of y.
The solution is (3, 6).
17. x + 3y = 11
x + 7y = 19
Substitute 4 for c in either equation to solve for .
SOLUTION: Because x and x have the same coefficients, subtract
the equations.
Angelina can buy 4 chocolate bars and 8 lollipops. So, B is the correct choice.
Now, substitute 2 for y in either equation to find the
value of x.
Use elimination to solve each system of
equations.
16. x + y = 9
x − y = −3
SOLUTION: Because y and −y have opposite coefficients, add the
equations.
The solution is (5, 2).
18. 9x − 24y = −6
3x + 4y = 10
SOLUTION: Multiply each term in the second equation by −3 to
eliminate the x coefficient.
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Now,
substitute
3 for
in either
value of y.
equation to find the
Page 7
Mid-Chapter
Quiz
The solution is (5, 2).
18. 9x − 24y = −6
3x + 4y = 10
SOLUTION: Multiply each term in the second equation by −3 to
eliminate the x coefficient.
The solution is (3, 2).
20. MULTIPLE CHOICE The Blue Mountain High
School Drama Club is selling tickets to their spring
musical. Adult tickets are $4 and student tickets are
$1. A total of 285 tickets are sold for $765. How
many of each type of ticket are sold?
F 145 adult, 140 student
Because 9x and −9x have opposite coefficients, add
the equations.
G 120 adult, 165 student
H 180 adult, 105 student
J 160 adult, 125 student
Now, substitute 1 for y in either equation to find the
value of y.
SOLUTION: Let a = the number of adult tickets sold and s = the
number of student tickets sold. So, a + s = 285 and
4a + 1s = 765.
Solve the first equation for s.
Substitute 285 – a for s in the second equation.
The solution is (2, 1).
19. −5x + 2y = −11
5x − 7y = 1
SOLUTION: Because −5x and 5x have opposite coefficients, add
the equations.
Use the solution for a in either equation to find the
value of s.
Now, substitute 2 for y in either equation to find the
value of x.
160 adult tickets and 125 student tickets were sold. So, J is the correct choice.
The solution is (3, 2).
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The
20. MULTIPLE CHOICE Blue Mountain High
School Drama Club is selling tickets to their spring
musical. Adult tickets are $4 and student tickets are
Page 8