Systems of Linear Equations Word Problems Worksheet | PDF - Free Printable

Let's solve each of the word problems from the Systems of Linear Equations Word Problems Worksheet step by step.

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Problem 1:

> A multiple choice test consists of 100 questions. A correct answer is worth 2 marks, while an incorrect answer is worth -1 mark. If a student receives a score of 80, how many questions did this student answer incorrectly?

#### Let:
- $ x $ = number of correct answers
- $ y $ = number of incorrect answers

We know:
1. Total questions: $ x + y = 100 $
2. Score: $ 2x - 1y = 80 $

Now solve the system:

From (1):
$ x = 100 - y $

Substitute into (2):
$ 2(100 - y) - y = 80 $
$ 200 - 2y - y = 80 $
$ 200 - 3y = 80 $
$ -3y = -120 $
$ y = 40 $

✔ Answer: The student answered 40 questions incorrectly.

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Problem 2:

> White chocolate costs $2.00 per bar, and dark chocolate costs $2.50 per bar. If you buy 15 bars of chocolate for $34 dollars, how many bars of dark chocolate did you buy?

Let:
- $ w $ = number of white chocolate bars
- $ d $ = number of dark chocolate bars

We have:
1. $ w + d = 15 $
2. $ 2w + 2.5d = 34 $

Solve:

From (1): $ w = 15 - d $

Substitute into (2):
$ 2(15 - d) + 2.5d = 34 $
$ 30 - 2d + 2.5d = 34 $
$ 30 + 0.5d = 34 $
$ 0.5d = 4 $
$ d = 8 $

✔ Answer: You bought 8 bars of dark chocolate.

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Problem 3:

> A cellular company’s revenue and cost functions are modeled by:
> $ R = 50n $, $ C = 10n + 300 $, where $ R $ is revenue, $ C $ is cost, and $ n $ is number of phones sold.
>
> At what point will this company have no profit? (Profit = Revenue – Cost)

No profit means:
$ R - C = 0 $

So:
$ 50n - (10n + 300) = 0 $
$ 50n - 10n - 300 = 0 $
$ 40n = 300 $
$ n = \frac{300}{40} = 7.5 $

But number of phones must be whole — however, since we're asked for the break-even point, it's acceptable to report the exact value.

✔ Answer: The company has no profit when 7.5 phones are sold.
But in real-world terms, they break even at 8 phones (since selling 7 gives loss, 8 gives profit). But mathematically, the break-even point is at $ n = 7.5 $.

So, the company breaks even at 7.5 units sold.

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Problem 4:

> The sum of two numbers is 17. The difference between the larger number and the smaller number is 7. What is the value of the smaller number?

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

Then:
1. $ x + y = 17 $
2. $ y - x = 7 $

Add equations:
$ (x + y) + (y - x) = 17 + 7 $
$ 2y = 24 $ → $ y = 12 $

Then $ x = 17 - 12 = 5 $

✔ Answer: The smaller number is 5.

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Problem 5:

> John has $5.05 in quarters and nickels in his pocket. If John only has 25 coins in his pocket, how many of the coins are quarters?

Let:
- $ q $ = number of quarters ($0.25 each)
- $ n $ = number of nickels ($0.05 each)

We have:
1. $ q + n = 25 $
2. $ 0.25q + 0.05n = 5.05 $

Multiply equation (2) by 100 to eliminate decimals:
$ 25q + 5n = 505 $

Divide entire equation by 5:
$ 5q + n = 101 $

Now use equation (1): $ n = 25 - q $

Substitute:
$ 5q + (25 - q) = 101 $
$ 5q + 25 - q = 101 $
$ 4q = 76 $
$ q = 19 $

✔ Answer: John has 19 quarters.

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Problem 6:

> A class of 32 consists of students who either have red or yellow shirts. If there are 12 more people with red shirts than there are people with yellow shirts, how many of the students have red shirts?

Let:
- $ r $ = number with red shirts
- $ y $ = number with yellow shirts

Then:
1. $ r + y = 32 $
2. $ r = y + 12 $

Substitute (2) into (1):
$ (y + 12) + y = 32 $
$ 2y + 12 = 32 $
$ 2y = 20 $
$ y = 10 $

Then $ r = 10 + 12 = 22 $

✔ Answer: 22 students have red shirts.

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✔ Final Answers Summary:

1. 40 questions answered incorrectly
2. 8 bars of dark chocolate
3. Break-even at 7.5 phones sold
4. Smaller number is 5
5. 19 quarters
6. 22 students have red shirts

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