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Solving Systems of Equations
In order to solve for two variables, you need to have two equations. If you only have one
equation there are an infinite amount of ordered pairs (x,y) that will work. For example:
4x – 2y = 16 you can have x = 4 and y = 0 (4,0) and (2, -2) and (0, -4) and an infinite amount of
others. To be able to solve for a single ordered pair, you need a second equation.
When we introduce the second equation, we will be able to solve for a single ordered pair that
will work in both equations. There are two ways to solve a system of equations (algebraically
and graphically). We will focus on solving algebraically. There are two methods of solving
algebraically (substitution and elimination). The key to both of them is changing one (or both)
equations so there is only one variable to solve for. Then you follow all the rules of solving for
the one variable. Then plug the value back into one of the original equations to find the value of
the second variable. Always state your answer as an ordered pair.
SUBSTITUTION
Example: x = 3y + 8
5x + 2y = 6
5(3y + 8) + 2y = 6 Substitute 3y +8 for the x in the 2nd equation
15y + 40 + 2y = 6 Distribute and solve.
17y + 40 = 6
17y = -34
y = -2
x = 3(-2) + 8
substitute the value for y back in to find x.
x= -6 + 8
x=2
(2, -2)
Check in BOTH ORIGINAL EQUATIONS!
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Solve each system and check (in both equations):
a) x = 2y + 1
c) 5x – y = 7
b) y = 3x + 4
5x – 6y = 13
9x + 2y = -37
d) x + 3y = 11
e) 7x + 9y = -74
6x – 5y = 20
4x + 2y = 28
f) 10x – y = 1
4x + y = -5
8x + 3y = -8
Solving Systems with Linear Combinations (“Elimination”):
Sometimes solving a system of equations using substitution can be very difficult. For these
problems we solve using Linear Combinations (or Elimination). With elimination you solve by
eliminating one of the variables. This is accomplished by adding the 2 equations together.
Before you can add the equations together, you need one of the two variables to have two things:
1) Same Coefficient
2) Different Signs (one positive and one negative)
When you add terms with the same coefficient and different signs, the term drops out. You then
solve for the variable that is left. After you have solved for one variable, you plug the value into
one of the original equations and solve for the 2nd variable (just like Substitution). Then, you
check the solution in both original equations. The only difference between Substitution and
Elimination is how you solve for the 1st variable. After that they are the same.
Examples:
A) Sometimes it works out that the 2 equations already have a variable with the same coefficient
and different signs. You can then just add the equations:
3x + 4y = 10 (The +4y and -4y cancel out
5x – 4y = -58 leaving you with just 8x.)
8x
= -48
8
8
x
= -6
Plug x = -6 in:
3(-6) + 4y = 10
-18 + 4y = 10
+18
+18
4y = 28
4
4
y=7
Final Solution: (-6, 7) CHECK IN BOTH!!!!
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B) Sometimes (usually) the equations do not have same coefficient and different signs, so we
have a little bit of manipulating to do.
3x + 8y = 25
With this system, nothing will drop out if we just add the
5x + 4y = 23
equations. So we will multiply the bottom one by (-2).
-2(5x + 4y = 23)
Now the y’s have the same coefficient with different signs.
- 10x -8y = -46
3x + 8y = 25
Now plug x = 3 in:
- 10x -8y = -46
3(3) + 8y = 25
- 7x
= -21
9 + 8y = 25
-7
-7
-9
-9
8y = 16
x
=3
8
8
y=2
Final Solution: (3,2) CHECK IN BOTH!!!!
C. Sometimes we need to manipulate both equations. We can do this by “criss crossing the
coefficients.”
6x + 7y = 11
This is different than Example B, because no coeffcient
5x – 6y = -50
goes into another evenly.
-5(6x + 7y = 11)
You need the negative sign to change the 6x to negative
6(5x – 6y = -50)
so the signs will be different.
You can also use 5 and -6. You can also “criss cross”
the y coefficients.
-30x – 35y = -55
30x – 36y = -300
- 71y = -355
-71
-71
y=5
Plug in y = 5
5x – 6(5) = -50
5x – 30 = -50
+30
+30
5x
= -20
5
5
x = -4
Final Solution: (-4, 5) CHECK IN BOTH!!!!
Practice:
1) 7x + 3y = 10
5x – 6y = 56
2) 11x + 5y = 27
4x + 6y = 60
3) 9x + 7y = 126
7x – 9y = -32
4) 12x – 5y = 63
8x + 3y = 23
5) 5x + 9y = 14
6x + 11y = 18
6) 10x – 9y = 36
4x + 3y = -12
7) 5x + 6y = 42
3x + 14y = 20
8) 7x – 5y = -42
8x + 3y = -48
9) 4x – 3y = 19
8x + 5y = 159
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Solve each system algebraically:
1) 5x - 2y = -9
7x + 2y = -27
2) -4x + 2y = -16
5x – 3y = 19
3) x = 2y -6
5y –3x = 11
4) 5x – 6y = -74
7x + 5y = 17
5) 4x – 5 = y
7x + 5y = 83
6) 7x + 4y = -11
5x + 2y = - 13
7) 5x – 6y = -17
3x + 8y = -16
8) x = 6 + 2y
6x – 5y = 15
9) 6x + 5y = 23
11x + 4y = 34
10) y = 3x + 4
8x – 9y = 59
11) 12x – 7y = 48
4x + 3y = -6
12) 9x – 4y = -88
2x + 5y = 4
13) 24x – 6y = -66
12x – 3y = -33
14) 5x – 6y = 42
15x – 18y = 54
15) 7x + 6y = -12
5x + 2y = -20
16) 13x – 3y = 78
4x + 6y = -66
17) 2y – 5 = x
4x – 11y = -38
18) 3x – 7y = -10
5x + 12y = -64
19) 6x – 17y = -104
4x – 7y = -39
20) 9x – 5y = -43
3x + 11y = 87
21) 9x = 11y + 25
5x – 12y = 8
22) 6y = 5x - 38
7x + 9y = 1
23) 6x + 5y = 33
5x + 37 = 3y
24) y = 3x + 5
12x – 7y = 1
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Answer Key to Algebraic Systems (page 19):
1) (-3,-3)
7) (-4, -.5)
13) many sol.
19) (2.5, 7)
2) (-7,-22)
8) (0,-3)
14) no sol.
20) (-1/3, 8)
3) (8,7)
9) (-2,7)
15) (-6,5)
21) (4,1)
4) (-4,9)
10) (-5,-11)
16) (3,-13)
22) (4,-3)
5) (4,11)
11) (1.5, -4)
17) (7,6)
23) (-2,9)
6) (-5,6)
12)(-8, 4)
18) (-8, -2)
24) (-4,-7)
1) 6x – 5y = -7
11x + 5y = 58
2) 5x + 4y = -69
5x -7y = 52
3) 6x + 7y = -28
5x – 14y = -182
4) 11x – 4y = 53
7x – 8y = 1
5) 3x – 7y = 42
2x + 5y = 57
6) 9x – 4y = 177
6x – 5y = 111
7) 8x – 11y = 77
6x + 4y = -28
8) 13x – 2y = 72
9x + 5y = -14
9) 12x = 20- 8y
5x – 6y = -57
10) 5y = 8x + 97
10x + 7y = 51
Answer Key for this sheet:
1) (3, 5)
2) (-5, -11)
3) (-14, 8)
4) (7,6)
5) (21,3)
6) (21, 3)
7) (0, -7)
8) (4, -10)
9) (-3,7)
10) (-4, 13)
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Word Problems
One Unknown, 2 Unknowns, Systems
1) Two trains leave the station at the same time. One is traveling east at a rate of 78 mph. The
other is traveling west at a rate of 72 mph. How long to they have to travel until they are 1,162.5
miles apart?
2) Two trains leave the station at the same time. One is traveling east at a rate of 69 mph. The
other is traveling east at a rate of 81 mph. How long to they have to travel until they are 174
miles apart? (Are they going in the same or different directions?)
3)
Two trains leave the station at the same time. One is traveling east at a speed of 78 mph.
The other is traveling west at 65 mph. How long until they are 1,515.8 miles apart?
4)
Two buses leave the station at the same time. One is traveling north at a speed of 73
mph. The other is traveling north at 61 mph. How long until they are 117 miles apart?
5)
Two planes leave the airport at the same time. One is traveling south at a speed of 158
mph. The other is traveling north at 185 mph. How long until they are 771.75 miles apart?
6) A truck driver is driving from Las Vegas to San Francisco. She drives the first 3 hours at a
rate of 57 mph. She then drives for 3 and a half hours at a rate of 68 mph. She finishes up the
trip by driving the last 4 and a half hours at a rate of 49 mph. What is her average rate of speed
for the entire trip?
7) A truck driver is driving from Boston to Chicago. She drives the first 3 hours at a rate of 67
mph. Then she hits some construction and is stuck driving the next two at a rate of 46 mph. She
then drives for 5 hours at a rate of 60 mph. She finishes up the trip by driving the last 3 hours at
a rate of 70 mph. What is her average rate of speed for the entire trip?
8)
A truck driver is driving from Chicago to Seattle. She drives 4 hours at 74 mph, 6 and a
half hours at 61 mph, 3 hours at 51 mph, and 2 and a half hours at 55 mph. What is her average
speed for the trip?
9)
A bus driver is driving from Indianapolis to Arkansas. He drives 5 hours at 73 mph, 5
and a half hours at 57 mph, 2 hours at 59 mph, and 1 and a half hours at 64 mph. What is his
average speed for the trip?
10)
A truck driver is driving from Cleveland to Dallas. She drives 3 hours at 61 mph, 4 and a
half hours at 70 mph, 5 hours at 49 mph, and 3 and a half hours at 65 mph. What is her average
speed for the trip?
11)
A jar of change has $78.15 in it. There are 7 more dimes than triple the amount of
nickels. There are 3 less quarters than double the amount of nickels. How many nickels, dimes,
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and quarters are in the jar? (There are no pennies.)
12) A jar of change has $209.25 in it. There are 10 dimes less than twice the amount of nickels
and 15 less quarters than triple the amount of nickels. How many nickels, dimes, and quarters
are in the jar? (There are no pennies.)
13) A jar of change has $57.25 in it. There are 2 dimes more than 3 times the amount of nickels
and 3 quarters less than double the amount of nickels. How many nickels, dimes, and quarters
are in the jar? (There are no pennies.)
14)
A jar of change has $12.40 in it. There are 5 less dimes than nickels. There are 3 less
quarters than twice the amount of nickels. How many nickels, dimes, and quarters are in the jar?
(There are no pennies.)
15)
A jar of change has $25.25 in it. There are 4 less nickels than twice the amount of dimes
and 3 quarters more than triple the amount of dimes. How many nickels, dimes, and quarters are
in the jar? (There are no pennies.)
16)
A store sells turkey sandwiches for $5.00 and ham sandwiches for $4.00. The store sells
5 more turkey sandwiches than double the amount of ham sandwiches and the total sales are
$1,089.00. How many of each were sold?
17)
A store sells hats for $12.50 and ties for $13.25. Last weekend the store sold 15 more
ties than 3 times the amount of hats and the sales were $9,290.25. How many of each were sold?
18)
A store sells workbooks for $8.50 and textbooks for $45.25. Last weekend the store sold
10 less textbooks than 3 times the amount of workbooks and the sales were $8,779.50. How
many of each were sold?
19)
A store sells cheeseburgers for $1.75 and bacon cheeseburgers for $2.25. Last weekend,
the store had 683.75 in total sales of cheeseburgers and bacon cheeseburgers. If there are 3 more
cheeseburgers sold than double the amount of bacon cheeseburgers, how many of each were
sold?
20) A snack stand at Yankee Stadium sells sodas for $4.25 and hot dogs for $5.50. During one
game the stand sold 26 more hot dogs than twice the amount of sodas. If the total sales for sodas
and hot dogs was $3,376; how many of each item were sold?
21) A deli sells quarts of milk for $1.09 and gallons for $2.49. The deli had 379.61 in total sales
of milk. If the deli sold 6 less gallons than double the amount of quarts, how many of each did
the deli sell?
22) A phone call cost $4.75. Introductory minutes cost $.22/min and additional minutes are
$.15/min. If there were 3 less additional minutes than double the introductory minutes, how
many minutes were billed at each rate?
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23) A phone call cost $3.36. Introductory minutes cost $.21/min and additional minutes are
$.09/min. If there were 6 less additional minutes than introductory minutes, how many
minutes were billed at each rate?
24) A 32-minute phone call cost $4.42. Introductory minutes cost $.17/min and additional
minutes are $.11/min. How many minutes were billed at each rate?
25) A 35-minute phone call cost $4.95. Introductory minutes cost $.16/min and additional
minutes are $.11/min. How many minutes were billed at each rate?
26) A phone call cost $4.74. Introductory minutes cost $.18/min and additional minutes are
$.13/min. If there were 6 more additional minutes than double the introductory minutes,
how long was the phone call?
27) A phone call cost $6.96. Introductory minutes cost $.21/min and additional minutes are
$.12/min. If there were 1 more additional minute than triple the introductory minutes,
how long was the phone call?
28) A 30-minute phone call cost $4.49. Introductory minutes cost $.19/minute and additional
minutes cost $.08/minute. How many minutes were billed at each rate?
29) Solid ties cost $21 and striped ties cost $24. The store sold 200 ties and made $4,413.
How many of each were sold?
30) At a movie theater adult tickets cost $9.00 and child tickets cost $4.00. 120 people
attended the last showing of Harry Potter and $720 was collected at the ticket booth.
How many of each ticket was sold?
31) Tiffany’s cell phone company’s daytime rate is $.20/minute and nighttime rate (after 7:00
PM) is $.12/minute. When she got her bill she saw a 29-minute phone call that cost
$4.12. When did the call begin? When did it end?
32) Senior citizens ride the bus for $.65. Adults ride the bus for $1.25. There were 65 people
on the bus and the bus driver collected $66.85. How many senior citizens were on the
bus?
33) Billy’s cell phone company’s daytime rate is $.15/minute and nighttime rate (after 8:00
PM) is $.11/minute. When he got his bill she saw a 32-minute phone call that cost $3.76.
When did the call begin? When did it end?
34) Two cars leave at 1:00 PM from the same place. One drives east at a rate of 71 mph and
the other drives west at 64 mph. At what time will they be 438.75 miles apart?
35) Two trains leave the train station at 12:45 PM, both traveling north. One is traveling at a
speed of 82 mph. The other is traveling at a speed of 105 mph. At what time will the
trains be 78.2 miles apart.
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36) The sum of two numbers is 57. The 2nd is 3 less than triple the 1st. Find both numbers.
37) Four numbers have a sum of 115. The 2nd number is 3 more than half the first. The 3rd
number is 7 less than double the first. The 4th number is 11 more than the first. Find all four
numbers.
38) Three numbers have the sum of 99. The 2nd number is 3 more than double the first. The 3rd
number is 3 more than the second. Find all three numbers.
39) The sum of 4 numbers is 162. The 1st number is 8 less than the 2nd. The 3rd number is 2
more than half the 2nd. The 4th number is 57 less than double the 2nd. What is the sum of the 2nd
and the 4th number?
40) The difference of 2 numbers is 21. The larger number is 4 more than double the smaller
number. Find both numbers.
41) The difference of 2 numbers is 25. The smaller number is 5 more than half the larger
number. Find both numbers.
42) Two thirds of a number is equal to 8 less than that same number. Find the number.
43) Triple a number is equal to the number increased by 38. Find the number.
44) Find 5 consecutive integers with the sum of 1,165.
45) Find 4 consecutive even integers such that the sum of the 2nd and 4th is -132.
46) Find 5 consecutive odd integers such that 3 times the first is 85 more than the sum of the 3rd
and the 5th.
47) Find five consecutive even integers with a sum of -210.
48) Find 6 consecutive odd integers such that the sum of the 1st and the 4th is 152.
49) Find 5 consecutive integers such that the sum of the 4th and the 5th is 56 more than triple the
first.
50) Find 5 consecutive odd integers such that 5 times the 2nd is 95 more than the sum of the 4th
and the 5th.
51) Find 5 consecutive even integers such that the sum of the 1st, 2nd, and 3rd is 134 less than 4
times the 5th.
52) The perimeter of a rectangle is 91inches. The length is 2 less than 4 times the width. What
are the dimensions of the rectangle?
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53) The perimeter of a triangle is 54 feet. The 2nd side is 9 more than half the 1st. The 3rd side is
5 more than the 1st. Find the three sides of the triangle
54) The area of a square is 144 in2. Find the perimeter.
55) In if there was a cube made from the dimensions in question 54. What would the volume
be?
56) The perimeter one side of a cube is 36 in. Find the area of one side and the volume of the
cube.
57) The perimeter of a rectangle is 57.5 inches. The length is 4 times the width. What are the
dimensions of the rectangle?
58) The perimeter of a triangle is 82 feet. The 2nd side is 12 more than one third of the 1st. The
3rd side is 50 less than the double the 1st. Find the three sides of the triangle
59) The perimeter of a rectangle is 96 inches. The width is 9 more than half the length. Find the
dimensions of the rectangle.
60) The area of a square is 196 in2. Find the perimeter.
61) In if there was a cube made from the dimensions in question 60. What would the
volume be?
62) The perimeter one side of a cube is 64 in. Find the area of one side and the volume of the
cube.
63) The perimeter of a rectangle is 72 inches. The length is 8 less than triple the width. What
are the dimensions of the rectangle?
64) The perimeter of a triangle is 92 feet. The 1st side is 12 inches more than half of the 2nd. The
3rd side is 11 inches less than the double the 2nd. Find the three sides of the triangle.
65) The perimeter if a rectangle is 80 cm. The length is 4 more than triple the width. Find the
area of the rectangle.
66) The perimeter of a rectangle is 94 inches. The width is 2 more than two-thirds the length.
Find the area of the rectangle.
67) The perimeter of a right triangle is 60 inches. The 2nd side is 4 more than double the 1st side.
The 3rd side is 4 less than triple the 1st side. Find the area of the triangle.
68) Two numbers have a sum of 67 and a difference of 41. Find the two numbers.
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69) Two numbers have a sum of 104 and a difference of 52. Find the quotient of the two
numbers.
70) Two numbers have a sum of 157 and a difference of 79. Find the quotient of the two
numbers.
71) Boohbah went into Dunkin Donuts for breakfast. Boohbah bought 5 donuts and 2 muffins
for $5.10. Boohbah went to order some more and the guy behind the counter made fun of him
for eating so much. He smacked around the guy behind the counter and bought 2 donuts and 7
muffins for $8.55. Find the price of 3 donuts and 4 muffins.
72) Boohbah was still angered by the guy behind the counter. So, he went and beat up some of
his family members. This made Boohbah hungry again. At Cherry Valley deli, Boohbah went
in and bought 7 TCS’s and 3 sodas for $50.15. This didn’t fill him up, so he went back in and
bought 2 more TCS’s and 2 more sodas for $16.10. What would the price of 4 TCS’s and 3
sodas be?
73) Boohbah finished off the clown from Dunkin Donuts, hid the body, and was then ready for
dessert, so he hit The Frozen Cup!! He bought 2 cones and a sundae for $9.70. Again, Boohbah
wanted more…. Much more…. So he bought 4 more cones and 5 more sundaes for $32.90. Find
the price of each item.
74) Triple a number decreased by a 2nd number is 263. The difference between the two numbers
is 41. Find quadruple the smaller number.
75) Mario went into the pizza place and bought 3 Sicilian slices and two regular slices and paid
12.65. Luigi followed and bought 4 Sicilian and 5 regular for $19.25. How much does 1 Sicilian
Answer Key:
Weighted Avg. Problems:
d = rt
1) 7 h 45 m
4) 9 h 45 m
2) 14 h 30 m
5) 2 h 15 m
3) 10h 36 m
Average speed:
6) 57.2 mph
9) 63.8 mph
7) 61.8 mph
10) 60.7 mph
8) 61.4 mph
Jar of Change:
11) 92 N, 283 D, 181 Q
12) 214 N, 418 D, 627 Q
34) t = 3 h 15 m
35) t = 3h 40m
4:15pm
4:25pm
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13) 68 N, 206 D, 133 Q
14) 21 N, 16 D, 39 Q
15) 48 N, 26 D, 81 Q
36) 15,42
37) 24,15,41,35
38) 18,39,42
39) 93 (4 #’s: 42,50,27,43)
Store Problems:
40) 17,38
16) 76 ham, 157 turkey
41) 35,60
17) 174 hats, 537 ties
42) 24
18) 64 wb, 182 tb
43) 19
19) 118 bcb, 239 cb
44) 231,232,233,234,235
20) 212 sodas, 450 hot dogs
45) -70,-68,-66,-64
21) 65 Quarts, 124 Gallons
46) 97,99,101,103,105
47) -46,-44,-42,-40,-38
Phone Call:
48) 73,75,77,79,81
22) 10 I, 17 A
49) -49,-48,-47,-46,-45
23) 13 I, 7 A
50) 33,35,37,39,41
24) 15 I, 17 A
51) 50,52,54,56,58
25) 22 I, 13 A
52) l=36 in , w= 9.5 in
26) 9 I, 24 A
53) 16,17,21
27) 12 I, 37 A
54) 48 in
28) 19 I, 11 A
55) 1,728 in3
56) A = 81 in2 V= 729 in3
Mixed: Remember the different Let:
57) l=23 in , w= 5.75 in
29) Let:
58) 36in,24in,22in
solid ties= x
59) l=26 in , w= 22 in
Striped ties= 200 – x
60) 56 in
29) 129 solid, 71 striped
61) 2,744 in3
30) 48 A, 72 C
62) A = 256 in2 V= 4,096 in3
31) 8 DT, 21 NT  call began at 6:52 and ended at 7:21
63) l=25 in , w= 11 in
32) 41 adults, 24 senior citizens
64) 25in,26in,41in
33) 6DT, 26 NT  call began at 7:54 and ended at 8:26
65) l=31 in , w= 9 in
66) l=27 in , w= 20 in
67) A = 120 in2 (sides= 10,24,26)
68) 54,13
69) 78, 26 quotient = 3
70) 118,39 product is 4,602
71)
72)
73)
74) 111,70 (280)
75) S =$2.75, R=2.20
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