Answer key for the "Solving Linear Equations (A)" math worksheet, featuring solutions to equations across six sections.

It looks like you've uploaded an image of a completed worksheet titled *"Solving Linear Equations (A)"* from Cazoom Maths, which includes answers to various linear equations. The worksheet is divided into sections (A through F), each containing different types of linear equations, increasing in complexity.

However, since this is the answer key, it doesn't contain the original problems — only the equations and their solutions. So, I can’t solve new problems unless you provide them.

But I can help you understand how to solve linear equations step-by-step using examples from this worksheet. Let’s go over the types of equations and how to solve them.

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🔍 Overview of the Sections

#### ✔ Section A: Basic addition/subtraction
Equations like:
- \( x + 3 = 11 \)
- \( x + 7 = 13 \)

How to solve:
Isolate \( x \) by subtracting the constant from both sides.

Example:
\[
x + 3 = 11 \\
x = 11 - 3 = 8
\]

✔ Answer: \( x = 8 \)

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#### ✔ Section B: Simple equations with constants on both sides
Like:
- \( 4 + x = 6 \)

Solution:
\[
x = 6 - 4 = 2
\]

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#### ✔ Section C: Subtraction equations
Like:
- \( x - 4 = 7 \)

Solution:
\[
x = 7 + 4 = 11
\]

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#### ✔ Section D: Multiplication equations
Like:
- \( 2x = 6 \)

Solution:
\[
x = \frac{6}{2} = 3
\]

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#### ✔ Section E: Fractional equations
Like:
- \( \frac{x}{3} = 4 \)

Solution:
Multiply both sides by 3:
\[
x = 4 \times 3 = 12
\]

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#### ✔ Section F: Mixed equations (more complex)
Examples:
- \( 12x - 19 = 30 \)
- \( \frac{x}{4} = 12 \)
- \( 5x + 100 = 20 \)

Let’s solve one:

##### Example: \( 12x - 19 = 30 \)

Step 1: Add 19 to both sides:
\[
12x = 30 + 19 = 49
\]

Step 2: Divide by 12:
\[
x = \frac{49}{12} \approx 4.08
\]

Wait! But the answer given is 49 — that suggests there might be a typo or misunderstanding.

Actually, let's check:
If \( 12x - 19 = 30 \), then:
\[
12x = 49 \Rightarrow x = \frac{49}{12}
\]
But the answer listed is 49 — which would mean they're solving for something else?

Wait — looking closely at the sheet:

> Problem 7: \( 12x - 19 = 30 \) → Answer: 49

That seems incorrect unless they’re asking for \( 12x \), not \( x \). But the instruction says "solve for \( x \)".

So likely, there’s a mistake in the answer key.

Let’s verify:
\[
12x - 19 = 30 \\
12x = 49 \\
x = \frac{49}{12} \approx 4.08
\]

But the answer sheet says 49, so maybe it's a formatting error — perhaps they meant \( x = 49 \)? That would make:
\[
12(49) - 19 = 588 - 19 = 569 \neq 30
\]

No, that’s way off.

Alternatively, maybe the equation was supposed to be:
\[
12x = 49 \Rightarrow x = \frac{49}{12}
\]

But no — the equation is clearly written as \( 12x - 19 = 30 \).

👉 Conclusion: The answer key has an error here. Correct answer should be \( x = \frac{49}{12} \), not 49.

Similarly, check other answers.

Let’s try another one:

> Problem 12: \( \frac{x}{11} = 11 \)

Answer: 121

✔ Correct!
\[
x = 11 \times 11 = 121
\]

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🚨 Potential Errors in Answer Key

Let’s check Section F, Problem 7 again:

> Equation: \( 12x - 19 = 30 \)
> Given answer: 49

But:
\[
12x = 30 + 19 = 49 \Rightarrow x = \frac{49}{12} \approx 4.08
\]

So the correct value of \( x \) is \( \frac{49}{12} \), but the answer sheet says 49, which is \( 12x \), not \( x \).

➡️ Likely, the answer key lists \( 12x \) instead of \( x \) — that’s a mistake.

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✔ Summary of How to Solve These

| Type | Example | Steps |
|------|--------|-------|
| Addition | \( x + 3 = 11 \) | Subtract 3: \( x = 8 \) |
| Subtraction | \( x - 4 = 7 \) | Add 4: \( x = 11 \) |
| Multiplication | \( 2x = 6 \) | Divide by 2: \( x = 3 \) |
| Division | \( \frac{x}{3} = 4 \) | Multiply by 3: \( x = 12 \) |
| Multi-step | \( 12x - 19 = 30 \) | Add 19 → divide by 12 |

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❗ Final Note

This appears to be an answer key, not a problem set. If you want me to solve specific equations from this sheet, please tell me which ones (e.g., "Solve Section F, Problem 7") and I’ll walk you through it correctly.

Also, note: There may be errors in the answer key — especially in Section F, Problem 7, where the answer should be \( \frac{49}{12} \), not 49.

Would you like me to:
- Re-solve all problems?
- Identify all errors?
- Explain how to check answers?

Let me know your goal!