Solving a linear equation using cross-multiplication and algebraic steps.

You've uploaded an image showing the step-by-step solution to the equation:

> Solve:
> \(\frac{x + 2}{3} = \frac{3x - 2}{6}\)

The image shows a correct solution, and I’ll walk you through each step with explanations so you understand *why* each step works.

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## ✔ Step 1: Cross Multiply

Equation:
\(\frac{x + 2}{3} = \frac{3x - 2}{6}\)

Action: Cross multiply → \(6(x + 2) = 3(3x - 2)\)

Why?
When two fractions are equal, you can cross-multiply to eliminate denominators. This is based on the property:

> If \(\frac{a}{b} = \frac{c}{d}\), then \(a \cdot d = b \cdot c\)

So here:
- Numerator of left × Denominator of right → \((x+2) \cdot 6\)
- Denominator of left × Numerator of right → \(3 \cdot (3x - 2)\)

✔ So: \(6(x + 2) = 3(3x - 2)\)

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## ✔ Step 2: Distribute

Left side: \(6(x + 2) = 6x + 12\)
Right side: \(3(3x - 2) = 9x - 6\)

Result:
\(6x + 12 = 9x - 6\)

Why?
Use the distributive property: \(a(b + c) = ab + ac\)

This simplifies both sides so we can combine like terms.

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## ✔ Step 3: Solve for x

We now solve the linear equation:

\(6x + 12 = 9x - 6\)

Goal: Get all x-terms on one side, constants on the other.

Subtract 6x from both sides:

→ \(12 = 3x - 6\)

Add 6 to both sides:

→ \(18 = 3x\)

Divide both sides by 3:

→ \(x = 6\)

✔ Final Answer: \(x = 6\)

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## ✔ Step 4: Check Your Solution

Plug \(x = 6\) back into the original equation to verify:

Left side: \(\frac{6 + 2}{3} = \frac{8}{3}\)
Right side: \(\frac{3(6) - 2}{6} = \frac{18 - 2}{6} = \frac{16}{6} = \frac{8}{3}\)

✔ Both sides equal \(\frac{8}{3}\), so the solution is correct!

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## 🧠 Summary

You solved the rational equation correctly by:

1. Cross multiplying to eliminate denominators.
2. Distributing to simplify both sides.
3. Isolating x using inverse operations.
4. Checking your answer by substitution — always good practice!

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## ✔ Final Answer:
\( \boxed{6} \)

Great job following the steps! The work shown in the image is accurate and well-explained.