Step-by-step algebraic problem-solving with handwritten notes and mathematical properties.

You've uploaded an image showing three algebraic proofs (Problems 12, 13, and 14) in a two-column format: Statements and Reasons. The goal is to solve each equation and justify each step using algebraic properties.

Let’s go through each problem carefully, correct any errors, and explain the reasoning.

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## 🔢 Problem 12: Given: \(\frac{2y - 1}{3} = 5\), Prove: \(y = 8\)

✔ Correct Solution:

| Statements | Reasons |
|----------------------------|-----------------------------|
| \(\frac{2y - 1}{3} = 5\) | Given |
| \(2y - 1 = 15\) | Multiplication Property of Equality (multiply both sides by 3) |
| \(2y = 16\) | Addition Property of Equality (add 1 to both sides) |
| \(y = 8\) | Division Property of Equality (divide both sides by 2) |

✔️ Explanation:
- Start with the given equation.
- Multiply both sides by 3 to eliminate the denominator.
- Add 1 to both sides to isolate the term with \(y\).
- Divide by 2 to solve for \(y\).

✔ Answer: \(y = 8\) — Correct!

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## 🔢 Problem 13: Given: \(2x + 20 = -4(2x - 2)\), Prove: \(x = -2\)

⚠️ Note: The handwritten solution says \(x = -1\), but that is incorrect. Let's solve it correctly.

✔ Correct Solution:

| Statements | Reasons |
|----------------------------|-----------------------------|
| \(2x + 20 = -4(2x - 2)\) | Given |
| \(2x + 20 = -8x + 8\) | Distributive Property |
| \(2x + 8x + 20 = 8\) | Addition Property of Equality (add 8x to both sides) |
| \(10x + 20 = 8\) | Combine like terms |
| \(10x = -12\) | Subtraction Property of Equality (subtract 20 from both sides) |
| \(x = -\frac{12}{10}\) | Division Property of Equality (divide by 10) |
| \(x = -\frac{6}{5}\) | Simplify fraction |

✘ Wait — this gives \(x = -\frac{6}{5}\), not \(-2\). But the problem says “Prove: \(x = -2\)”. That suggests there may be a typo in the original problem.

Let’s check if \(x = -2\) actually satisfies the original equation:

Left side: \(2(-2) + 20 = -4 + 20 = 16\)

Right side: \(-4(2(-2) - 2) = -4(-4 - 2) = -4(-6) = 24\)

→ 16 ≠ 24 → So \(x = -2\) is NOT a solution.

But maybe the original equation was meant to be:

> \(2x + 20 = -4(2x + 2)\)

Let’s test that:

Left: \(2x + 20\)

Right: \(-4(2x + 2) = -8x - 8\)

Set equal:
\(2x + 20 = -8x - 8\)
Add 8x: \(10x + 20 = -8\)
Subtract 20: \(10x = -28\) → \(x = -2.8\) — still not -2.

What if it was:
> \(2x + 20 = -4(x - 2)\)

Then:
Right: \(-4x + 8\)
Equation: \(2x + 20 = -4x + 8\)
Add 4x: \(6x + 20 = 8\)
Subtract 20: \(6x = -12\) → \(x = -2\) ✔

So likely, the original problem had a typo, and should have been:

> Given: \(2x + 20 = -4(x - 2)\), Prove: \(x = -2\)

✔ Corrected Proof:

| Statements | Reasons |
|----------------------------|-----------------------------|
| \(2x + 20 = -4(x - 2)\) | Given (corrected) |
| \(2x + 20 = -4x + 8\) | Distributive Property |
| \(2x + 4x + 20 = 8\) | Addition Property (add 4x) |
| \(6x + 20 = 8\) | Combine like terms |
| \(6x = -12\) | Subtraction Property |
| \(x = -2\) | Division Property |

✔️ Answer: \(x = -2\) — Now correct!

➡️ Conclusion for Problem 13: The handwritten solution has an error — it says \(x = -1\), which is wrong. Also, the original equation as written does *not* yield \(x = -2\). There’s likely a typo in the problem statement. Assuming the intended equation was \(2x + 20 = -4(x - 2)\), then \(x = -2\) is correct.

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## 🔢 Problem 14: Given: \(18x - 2(3x - 2) = 5x - 16\), Prove: \(x = -2\)

✔ Correct Solution:

| Statements | Reasons |
|----------------------------|-----------------------------|
| \(18x - 2(3x - 2) = 5x - 16\) | Given |
| \(18x - 6x + 4 = 5x - 16\) | Distributive Property |
| \(12x + 4 = 5x - 16\) | Combine like terms |
| \(12x - 5x + 4 = -16\) | Subtraction Property (subtract 5x) |
| \(7x + 4 = -16\) | Combine like terms |
| \(7x = -20\) | Subtraction Property (subtract 4) |
| \(x = -\frac{20}{7}\) | Division Property |

✘ This gives \(x = -\frac{20}{7}\), not \(-2\).

Check if \(x = -2\) satisfies the original equation:

Left: \(18(-2) - 2(3(-2) - 2) = -36 - 2(-6 - 2) = -36 - 2(-8) = -36 + 16 = -20\)

Right: \(5(-2) - 16 = -10 - 16 = -26\)

→ -20 ≠ -26 → So \(x = -2\) is NOT a solution.

Wait — perhaps the equation was meant to be:

> \(18x - 2(3x - 2) = 5x + 16\) ← positive 16?

Try that:

Left: same → -20
Right: \(5(-2) + 16 = -10 + 16 = 6\) → still not equal.

What if it was:

> \(18x - 2(3x - 2) = 5x - 20\)

Left: -20
Right: \(5(-2) - 20 = -10 - 20 = -30\) → no.

Alternatively, maybe the left side is different.

Let’s assume the goal is to prove \(x = -2\), so let’s work backward.

If \(x = -2\), plug into left side:

\(18(-2) - 2(3(-2) - 2) = -36 - 2(-6 - 2) = -36 + 16 = -20\)

So right side must also be -20 → so equation should be:

> \(18x - 2(3x - 2) = 5x - 20\)

Let’s solve that:

Left: \(18x - 6x + 4 = 12x + 4\)

Set equal: \(12x + 4 = 5x - 20\)

Subtract 5x: \(7x + 4 = -20\)

Subtract 4: \(7x = -24\) → \(x = -\frac{24}{7}\) — still not -2.

Wait — what if the equation was:

> \(18x - 2(3x + 2) = 5x - 16\)

Left: \(18x - 6x - 4 = 12x - 4\)

Set equal: \(12x - 4 = 5x - 16\)

Subtract 5x: \(7x - 4 = -16\)

Add 4: \(7x = -12\) → \(x = -\frac{12}{7}\)

Still not -2.

Let’s try:

> \(18x - 2(3x - 2) = 5x - 12\)

Left: 12x + 4
Set equal: 12x + 4 = 5x - 12
7x = -16 → x = -16/7

Not working.

Maybe the original problem is correct, and the “prove x = -2” is wrong.

Let’s solve the original as written:

Given: \(18x - 2(3x - 2) = 5x - 16\)

Step-by-step:

1. Distribute: \(18x - 6x + 4 = 5x - 16\) → \(12x + 4 = 5x - 16\)

2. Subtract 5x: \(7x + 4 = -16\)

3. Subtract 4: \(7x = -20\)

4. Divide: \(x = -\frac{20}{7}\)

✔ So the correct solution is \(x = -\frac{20}{7}\), not \(-2\).

The handwritten proof shows:

> \(18x - 2(3x - 2) = 5x - 16\)
> → \(18x - 6x + 2 = 5x - 16\) ← ✘ ERROR HERE! It should be +4, not +2.

That’s the mistake! They distributed \(-2 \times (-2)\) as +2, but it’s +4.

Then they got:

> \(12x + 2 = 5x - 16\)
> → \(7x + 2 = -16\)
> → \(7x = -18\)
> → \(x = -\frac{18}{7}\) — still not -2.

Then they wrote \(x = -2\), which is completely wrong.

✔ Correct Answer: \(x = -\frac{20}{7}\)

➡️ Conclusion for Problem 14: The handwritten solution contains a distribution error (+2 instead of +4) and incorrectly concludes \(x = -2\). The actual solution is \(x = -\frac{20}{7}\).

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## 🧾 Final Summary:

| Problem | Given Equation | Claimed Answer | Actual Answer | Verdict |
|---------|----------------------------------|----------------|-------------------|---------------------|
| 12 | \(\frac{2y - 1}{3} = 5\) | \(y = 8\) | ✔ \(y = 8\) | ✔️ Correct |
| 13 | \(2x + 20 = -4(2x - 2)\) | \(x = -2\) | ✘ \(x = -\frac{6}{5}\) | ⚠️ Likely typo; if corrected to \(-4(x - 2)\), then \(x = -2\) |
| 14 | \(18x - 2(3x - 2) = 5x - 16\) | \(x = -2\) | ✘ \(x = -\frac{20}{7}\) | ✘ Incorrect due to distribution error |

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## ✔ Recommendations:

- For Problem 13: Confirm the original equation. If it’s indeed \(-4(2x - 2)\), then \(x = -\frac{6}{5}\); if it’s \(-4(x - 2)\), then \(x = -2\).
- For Problem 14: Fix the distribution: \(-2 \times (-2) = +4\), not +2. Then solve correctly to get \(x = -\frac{20}{7}\).

Let me know if you want to adjust the equations or re-solve with corrections!