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Unit 10: Systems of Linear Equations/Inequalities
Study Guide
Knowledge and Understanding Math
1. What are the possible solutions for a system of equations?
A system of linear equations can have one solution, no solutions, or many solutions.
2. What are all methods you can use to solve a system of equations?
Table
Graph
Substitution
Linear Combinations
Set Equal
Multiply 1st
3. Is it possible for a system of inequalities to have no solution? Explain.
is possible of a system of linear inequalities to have no solution. If the two equations are parallel
and are shaded away from each other the inequality has no solution.
4. Is (8, 5) the solution to this system of equations?
5x – 4y = 20
3y = 2x + 1
5x – 4y = 20
3y = 2x + 1
5(8) – 4(5) = 20
3(5) = 2(8) + 1
40 – 20 = 20
15 = 16 + 1
20 = 20
15 = 17
True
False
4. Not a Solution
5. Graph the system of equations. From the graph, identify the solution.
x>2
y<3
6. Solve this system of equations using any method you choose:
5y + 2x = 5x + 1
3x – 2y= 3 + 3y
5y + 2x = 5x + 1
- 5x – 5x
5y – 3x = 1
-3x + 5y = 1
3x – 2y = 3 + 3y
- 3y
- 3y
3x – 5y = 3
-3x + 5y = 1
3x – 5y = 3
0=4
No Solution
6. No Solution
7. Is (4, -2) a solution to the system of inequalities:
x<5
x + 2y > 1
x<5
x + 2y > 1
4<5
4 + 2(-2) > 1
True
4–4>1
0>1
False
7. Not a Solution
8. Graph this system of inequalities:
y>x–3
m=1
b = -3
x+y<2
-x
-x
y < -x + 2
m = -1
b=2
What is the solution? Anything in gray
9. You are painting the white lines around the perimeter of a tennis court. You measure and find that
the perimeter is 228 feet and the length is 42 feet longer than the width. Write and solve a
system of linear equations to find the length and width of the tennis court. Let w be the width of
the tennis court and let ℓ be the length of the tennis court.
228 = 2L + 2W
2L + 2W = 228
L - 42 = W
L - 42 = W
2L + 2(L - 42) = 228
78 - 42 = W
2L + 2L - 84 = 228
36 = W
4L - 84 = 228
+ 84 + 84
4L = 312
L = 78
10. In an academic competition, scoring is based on a written examination and an oral presentation.
The written examination cannot exceed 65 points and the oral presentation cannot exceed 35
points. Write and graph a system of inequalities for the school team can receive. Find 1 possible
solution.
x = Written
y = Oral
x < 65
y < 35
Graph counted by 10’s. No negatives are shaded
since negative points cannot be earned.
11. Solve the system of equations:
5x = 1 – 3y
+ 3y + 3y
5x + 3y = 1
2x + 9y = 16
-3(5x + 3y = 1)
2x + 9y = 16
5x = 1 – 3y
2x + 9y = 16
-15x – 9y = -3
-13x = 13
x = -1
2x + 9y = 16
2(-1) + 9y = 16
-2 + 9y = 16
+2
+2
9y = 18
y=2
12. Tickets for a school play cost $4 for adults and $2 for students. At the end of the play, the
school sold a total of 105 tickets and collect $360.
a. Write a linear system. Let x be the number of adult tickets sold and let y be the number of
student tickets sold.
4x + 2y = 360
x + y = 105
b. Find the number of adult tickets sold and the number of student tickets sold.
4x + 2y = 360
4x + 2y = 360
x + y = 105
-2(x + y) = 105
-2x -2y = -210
75 + y = 105
2x = 150
-75
-75
x = 75
y = 30
75 adult tickets and 30 student tickets
13. Write an equation for each cookie and cake then solve the system. After how many hours does the
number of cookies equal the number of cakes? How many cookies and cakes will you have?
x = hours
y = total
y = 1000x + 15500
y = 250x + 38000
1000x + 15500 = 250x +38000
- 250x
- 250x
750x + 15500 = 38000
- 15500 - 15500
750x = 22500
x = 30 hours
Sweet Number
Started
with
Cookie
15,500
Cake
38,000
Baked per
hour
1000
250
y = 1000x + 15500
y = 1000(30) + 15500
y = 30,000 + 15,500
y = 45,500
Cookies = 1000(30) + 15,500
Cakes = 250(30) + 38000
Cookies = 30,000 + 15,500
Cakes = 7,500 + 38000
Cookies = 45,500
Cakes = 45,500
14. A hotel rents a double-occupancy room for $20 more than single-occupancy room. One night,
the hotel took in $3115 after renting 15 double-occupancy rooms and 26 single-occupancy
rooms. Write and solve a linear system to find the cost of renting a double-occupancy room and
the cost of renting a single-occupancy room.
x = single occupancy room
26x + 15(x + 20) = 3115
x + 20 = y
y = double occupancy room
26x + 15x + 300 = 3115
68.66 + 20 = y
x + 20 = y
41x + 300 = 3115
$88.66 = y
26x + 15y = 3115
41x = 2815
x = $68.66
15. For a system of linear equations to have no solutions what must be true?
The system of linear equations must be parallel (have the same slope) in order to have no solution.
16. Describe the steps needed to solve the linear system below.
1.
2.
3.
4.
5.
3x - 5y = 8
2x + 4y = 6
Multiply the first equation through by 4.
Multiply the second equation through by 5.
Add the equations together causing the y’s to be eliminated.
Solve for x.
Plug your solution into one of the original equations and solve for y.
17. What would be the value of y in the first equation if you choose to solve the system of equations
below by substitution?
4x – 2y = -3
8x + 6y = 48
4x – 2y = -3
- 4x
- 4x
-2y = -4x – 3
÷-2
÷ -2
y = 2x +3/2
20. A website allows users to download individual songs or an entire album. All individual songs cost
the same to download, and all albums cost the same to download. Ryan pays $14.94 to download 5
individual songs and 1 album. Seth pays $22.95 to download 3 individual songs and 2 albums. How
much does the website charge to download 7 individual songs and 3 albums?
x = songs
-2(5x + 1y = 14.94)
-10x – 2y = -29.88
5x + y = 14.94
y = albums
3x + 2y = 22.95
3x + 2y = 22.95
5(.99) + y = 14.94
5x + 1y = 14.94
-7x = -6.93
4.95 + y = 14.94
3x + 2y = 22.95
x = 0.99
y = 9.99
7x + 3y = ?
7(.99) + 3(9.99) =
6.93 + 29.97 =
$36.90