Teacher's answer key for a worksheet on systems of linear equations word problems, including problems on test scoring, chocolate bar purchases, and cellular company profit.

The image you've uploaded is a "Systems of Linear Equations Word Problems Worksheet" with a Teacher's Answer Key. It contains three word problems, each solved using systems of linear equations. Below is a detailed explanation and solution for each problem, as shown in the worksheet.

---

Problem 1: Multiple Choice Test

> 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 30, how many questions did this student answer incorrectly?

#### Step-by-step Solution:

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

We know:
1. Total questions: $ x + y = 100 $
2. Scoring: Each correct = +2, Incorrect = -1 → Total score = $ 2y - x = 30 $

So we have the system:
$$
\begin{align*}
x + y &= 100 \quad \text{(Equation 1)} \\
2y - x &= 30 \quad \text{(Equation 2)}
\end{align*}
$$

Now solve:

From Equation 1: $ x = 100 - y $

Substitute into Equation 2:
$$
2y - (100 - y) = 30 \\
2y - 100 + y = 30 \\
3y = 130 \\
y = \frac{130}{3} \approx 43.33
$$

Wait! This gives a non-integer value — but that can't be right since number of questions must be whole numbers.

Wait — let’s double-check the scoring.

Actually, the score is 30, and:
- Correct: +2
- Incorrect: -1

So total score: $ 2y - x = 30 $

And $ x + y = 100 $

Let’s solve again:

From $ x + y = 100 $ → $ x = 100 - y $

Plug into score equation:
$$
2y - (100 - y) = 30 \\
2y - 100 + y = 30 \\
3y = 130 \\
y = \frac{130}{3} \approx 43.33
$$

Still not integer. But the answer key says $ y = 60 $, $ x = 40 $.

Let’s check if that works:

- Correct: 60 → $ 60 \times 2 = 120 $
- Incorrect: 40 → $ 40 \times (-1) = -40 $
- Total: $ 120 - 40 = 80 $

But the student scored 30, not 80.

Wait — there's a mistake in the worksheet.

The answer key says:
> So $ y = 60 $ and $ x = 40 $

But then:
- Score = $ 2(60) - 1(40) = 120 - 40 = 80 $

But the problem says the student got 30, not 80.

So either:
- The problem says score is 80, or
- The answer is wrong.

But the worksheet says: "If a student receives a score of 30", and then says “So $ y = 60 $, $ x = 40 $” — which gives 80, not 30.

This is inconsistent.

Let’s fix it.

We want:
$$
x + y = 100 \\
2y - x = 30
$$

Solve:
From first: $ x = 100 - y $

Substitute:
$$
2y - (100 - y) = 30 \\
2y - 100 + y = 30 \\
3y = 130 \\
y = \frac{130}{3} \approx 43.33
$$

Not possible.

So perhaps the score is 80, not 30?

Try: $ 2y - x = 80 $, $ x + y = 100 $

Then:
$ x = 100 - y $

$ 2y - (100 - y) = 80 $

$ 2y - 100 + y = 80 $

$ 3y = 180 $

$ y = 60 $, $ x = 40 $

Yes! That matches the answer key.

So likely, the problem was meant to say:
> "If a student receives a score of 80, how many questions did this student answer incorrectly?"

Then the answer is 40 incorrect.

✔ So the problem has a typo: score should be 80, not 30.

---

✔ Corrected Problem 1:

> A multiple choice test has 100 questions. Correct = +2, Incorrect = -1. If a student scores 80, how many were answered incorrectly?

Let:
- $ x $ = incorrect
- $ y $ = correct

$$
\begin{align*}
x + y &= 100 \\
2y - x &= 80
\end{align*}
$$

Add equations:
$$
(x + y) + (2y - x) = 100 + 80 \\
3y = 180 \Rightarrow y = 60 \\
x = 100 - 60 = 40
$$

✔ So 40 questions were answered incorrectly.

---

Problem 2: Chocolate Bars

> White chocolate costs \$2.00 per bar, dark chocolate \$2.50 per bar. You buy 15 bars for \$34. How many dark chocolate bars did you buy?

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

Equations:
$$
\begin{align*}
w + d &= 15 \quad \text{(total bars)} \\
2w + 2.5d &= 34 \quad \text{(total cost)}
\end{align*}
$$

Solve:

From first: $ w = 15 - d $

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

Then $ w = 15 - 8 = 7 $

✔ So you bought 8 dark chocolate bars.

Answer key is correct.

---

Problem 3: Profit Break-even

> Revenue: $ R = 50n $
> Cost: $ C = 20n + 300 $
> Find $ n $ such that profit = 0 → Revenue = Cost

Set:
$$
50n = 20n + 300 \\
50n - 20n = 300 \\
30n = 300 \\
n = 10
$$

✔ So the company must sell 10 phones to break even.

Answer key is correct.

---

✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | 40 incorrect answers *(assuming score is 80, not 30)* |
| 2 | 8 dark chocolate bars |
| 3 | 10 phones sold |

---

❗Note:

There is a typo in Problem 1 — the score should be 80, not 30, for the given answer to be correct. Otherwise, no integer solution exists.

Let me know if you'd like a corrected version of the worksheet!