Free Systems of Equations Word Problems Worksheet Collection - Free Printable

Let's solve each of these systems of equations word problems step by step.

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1) Movie Tickets

- Adult ticket: $5
- Student ticket: $3
- Total tickets: 18
- Total cost: $82

Let:
- $ a $ = number of adult tickets
- $ s $ = number of student tickets

Equations:
$$
a + s = 18 \quad \text{(1)}
$$
$$
5a + 3s = 82 \quad \text{(2)}
$$

From (1): $ s = 18 - a $

Substitute into (2):
$$
5a + 3(18 - a) = 82 \\
5a + 54 - 3a = 82 \\
2a = 28 \\
a = 14
$$

✔ Answer: 14 adult tickets

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2) Eva and Nicole’s Purchase

Let:
- $ x $ = price of one shirt
- $ y $ = price of one hat

Eva: 2 shirts, 5 hats → $ 2x + 5y = 154 $
Nicole: 3 shirts, 4 hats → $ 3x + 4y = 168 $

System:
$$
2x + 5y = 154 \quad \text{(1)} \\
3x + 4y = 168 \quad \text{(2)}
$$

Multiply (1) by 3: $ 6x + 15y = 462 $
Multiply (2) by 2: $ 6x + 8y = 336 $

Subtract:
$$
(6x + 15y) - (6x + 8y) = 462 - 336 \\
7y = 126 \Rightarrow y = 18
$$

Plug back into (1):
$$
2x + 5(18) = 154 \\
2x + 90 = 154 \\
2x = 64 \Rightarrow x = 32
$$

✔ Answer: $32 per shirt

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3) Pigs and Ducks

- Total animals: 10
- Total legs: 36
- Pigs have 4 legs, ducks have 2 legs

Let:
- $ p $ = number of pigs
- $ d $ = number of ducks

$$
p + d = 10 \quad \text{(1)} \\
4p + 2d = 36 \quad \text{(2)}
$$

From (1): $ d = 10 - p $

Substitute into (2):
$$
4p + 2(10 - p) = 36 \\
4p + 20 - 2p = 36 \\
2p = 16 \Rightarrow p = 8
$$

✔ Answer: 8 pigs

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4) Field Trip Vehicles

- Total students: 195
- Total vehicles: 19
- Car holds 5 students
- Bus holds 25 students

Let:
- $ c $ = number of cars
- $ b $ = number of buses

$$
c + b = 19 \quad \text{(1)} \\
5c + 25b = 195 \quad \text{(2)}
$$

Simplify (2): divide by 5 → $ c + 5b = 39 $

Now subtract (1) from this:
$$
(c + 5b) - (c + b) = 39 - 19 \\
4b = 20 \Rightarrow b = 5
$$

✔ Answer: 5 buses

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5) Theater Tickets

Let:
- $ s $ = price of senior ticket
- $ c $ = price of child ticket

Mr. Smith: $ 8s + 5c = 136 $
Mr. Jackson: $ 4s + 6c = 96 $

System:
$$
8s + 5c = 136 \quad \text{(1)} \\
4s + 6c = 96 \quad \text{(2)}
$$

Multiply (2) by 2: $ 8s + 12c = 192 $

Subtract (1):
$$
(8s + 12c) - (8s + 5c) = 192 - 136 \\
7c = 56 \Rightarrow c = 8
$$

Plug into (2):
$$
4s + 6(8) = 96 \\
4s + 48 = 96 \\
4s = 48 \Rightarrow s = 12
$$

✔ Answer: $12 per senior ticket

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6) Two Numbers – Difference 6, Sum 14

Let:
- $ x $ = bigger number
- $ y $ = smaller number

$$
x - y = 6 \quad \text{(1)} \\
x + y = 14 \quad \text{(2)}
$$

Add equations:
$$
2x = 20 \Rightarrow x = 10
$$

✔ Answer: 10

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7) Two-Digit Number – Digit Sum is 7, Reverse Increases by 9

Let the number be $ 10a + b $, where:
- $ a $ = tens digit
- $ b $ = units digit

Given:
- $ a + b = 7 $
- Reversed number: $ 10b + a $
- $ 10b + a = 10a + b + 9 $

Simplify second equation:
$$
10b + a - 10a - b = 9 \\
9b - 9a = 9 \Rightarrow b - a = 1
$$

Now:
$$
a + b = 7 \\
b - a = 1
$$

Add:
$$
2b = 8 \Rightarrow b = 4 \Rightarrow a = 3
$$

Number: $ 10a + b = 30 + 4 = 34 $

Check: reverse is 43; $ 43 - 34 = 9 $ ✔

✔ Answer: 34

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8) Two Numbers – Difference 18, Sum 66

Let:
- $ x $ = larger
- $ y $ = smaller

$$
x - y = 18 \\
x + y = 66
$$

Add:
$$
2x = 84 \Rightarrow x = 42 \Rightarrow y = 24
$$

✔ Answer: 42 and 24

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9) Rectangle Dimensions

Let:
- $ w $ = width
- $ l $ = length

Given:
- $ l = 2w + 3 $
- Perimeter = 30 → $ 2(l + w) = 30 \Rightarrow l + w = 15 $

Substitute:
$$
(2w + 3) + w = 15 \\
3w + 3 = 15 \\
3w = 12 \Rightarrow w = 4 \\
l = 2(4) + 3 = 11
$$

✔ Answer: Length is 11 meters

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10) Nickels and Dimes

- Nickels = $0.05
- Dimes = $0.10
- Total coins: 44
- Total value: $2.95

Let:
- $ n $ = number of nickels
- $ d $ = number of dimes

$$
n + d = 44 \quad \text{(1)} \\
0.05n + 0.10d = 2.95 \quad \text{(2)}
$$

Multiply (2) by 100 to eliminate decimals:
$$
5n + 10d = 295
$$

Divide by 5:
$$
n + 2d = 59 \quad \text{(2')}
$$

Now subtract (1):
$$
(n + 2d) - (n + d) = 59 - 44 \\
d = 15
\Rightarrow n = 44 - 15 = 29
$$

✔ Answer: 29 nickels

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✔ Final Answers:

1) 14
2) $32
3) 8 pigs
4) 5 buses
5) $12
6) 10
7) 34
8) 42 and 24
9) 11 meters
10) 29 nickels

Let me know if you'd like explanations in a different format or want to see the steps graphically!