Systems of Linear Equations Worksheets with Answer Key - Free Printable

Let's solve each problem step by step using systems of linear equations.

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Problem 1: Boat and Current

> A boat traveled 336 miles downstream and back. The trip downstream took 12 hours. The trip back took 14 hours. Find the speed of the boat in still water. What is the speed of the current?

#### Step 1: Define variables
- Let $ b $ = speed of the boat in still water (in mph)
- Let $ c $ = speed of the current (in mph)

Then:
- Downstream speed = $ b + c $
- Upstream speed = $ b - c $

#### Step 2: Use distance = rate × time

Downstream:
$ 336 = (b + c) \cdot 12 $ → Equation (1):
$$
12(b + c) = 336
$$

Upstream:
$ 336 = (b - c) \cdot 14 $ → Equation (2):
$$
14(b - c) = 336
$$

#### Step 3: Solve the system

Divide both sides of (1) by 12:
$$
b + c = 28 \quad \text{(1a)}
$$

Divide both sides of (2) by 14:
$$
b - c = 24 \quad \text{(2a)}
$$

Now add (1a) and (2a):
$$
(b + c) + (b - c) = 28 + 24 \\
2b = 52 \Rightarrow b = 26
$$

Substitute $ b = 26 $ into (1a):
$$
26 + c = 28 \Rightarrow c = 2
$$

#### ✔ Answer:
- Speed of boat in still water: 26 mph
- Speed of current: 2 mph

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Problem 2: Umbrellas and Shoes

> Michael bought 2 umbrellas and 3 pairs of shoes for $5.13. He then bought another umbrella and another 2 pairs of shoes for $3.09. Find the cost of each umbrella and each pair of shoes.

#### Step 1: Define variables
- Let $ u $ = cost of one umbrella
- Let $ s $ = cost of one pair of shoes

#### Step 2: Write equations

First purchase:
$$
2u + 3s = 5.13 \quad \text{(1)}
$$

Second purchase:
$$
u + 2s = 3.09 \quad \text{(2)}
$$

#### Step 3: Solve the system

Multiply equation (2) by 2:
$$
2u + 4s = 6.18 \quad \text{(2a)}
$$

Now subtract equation (1) from (2a):
$$
(2u + 4s) - (2u + 3s) = 6.18 - 5.13 \\
s = 1.05
$$

Now plug $ s = 1.05 $ into equation (2):
$$
u + 2(1.05) = 3.09 \\
u + 2.10 = 3.09 \\
u = 0.99
$$

#### ✔ Answer:
- Cost of one umbrella: $0.99
- Cost of one pair of shoes: $1.05

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Problem 3: Fraction Word Problem

> The denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are decreased by 6, then the denominator becomes 12 times the numerator. Determine the fraction.

#### Step 1: Define variables
- Let $ x $ = numerator
- Then denominator = $ 2x + 4 $

So original fraction: $ \frac{x}{2x + 4} $

After decreasing both by 6:
- New numerator: $ x - 6 $
- New denominator: $ (2x + 4) - 6 = 2x - 2 $

Given: New denominator = 12 × new numerator:
$$
2x - 2 = 12(x - 6)
$$

#### Step 2: Solve the equation

$$
2x - 2 = 12x - 72 \\
-2 + 72 = 12x - 2x \\
70 = 10x \Rightarrow x = 7
$$

Now find denominator:
$$
2x + 4 = 2(7) + 4 = 14 + 4 = 18
$$

So the fraction is $ \frac{7}{18} $

#### ✔ Answer:
The fraction is $ \frac{7}{18} $

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Problem 4: Two Numbers – Difference and Sum

> The difference of two numbers is 3. Their sum is 13. Find the numbers.

#### Step 1: Define variables
- Let $ x $ and $ y $ be the two numbers

Equations:
$$
x - y = 3 \quad \text{(1)} \\
x + y = 13 \quad \text{(2)}
$$

#### Step 2: Add equations
$$
(x - y) + (x + y) = 3 + 13 \\
2x = 16 \Rightarrow x = 8
$$

Plug into (2):
$$
8 + y = 13 \Rightarrow y = 5
$$

#### ✔ Answer:
The numbers are 8 and 5

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Problem 5: Two Numbers – Sum and Difference

> Find the value of two numbers if their sum is 16 and their difference is 6.

#### Step 1: Define variables
- Let $ x $ and $ y $ be the numbers

Equations:
$$
x + y = 16 \quad \text{(1)} \\
x - y = 6 \quad \text{(2)}
$$

#### Step 2: Add equations
$$
(x + y) + (x - y) = 16 + 6 \\
2x = 22 \Rightarrow x = 11
$$

Plug into (1):
$$
11 + y = 16 \Rightarrow y = 5
$$

#### ✔ Answer:
The numbers are 11 and 5

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✔ Final Answers Summary:

1. Boat:
- Speed in still water: 26 mph
- Speed of current: 2 mph

2. Umbrella and Shoes:
- Umbrella: $0.99
- Shoes: $1.05

3. Fraction:
- $ \frac{7}{18} $

4. Numbers with sum 13 and difference 3:
- 8 and 5

5. Numbers with sum 16 and difference 6:
- 11 and 5

Let me know if you'd like these problems explained visually or with graphs!