Question: Solve the equation.
$\frac{4}{7} (x + 14) = \frac{2}{3} (x - 7)$
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The solution to the equation is $x = \square$.
1. **State the problem:** Solve the equation $$\frac{4}{7} (x + 14) = \frac{2}{3} (x - 7)$$.
2. **Use the distributive property:** Multiply each fraction by the terms inside the parentheses.
$$\frac{4}{7}x + \frac{4}{7} \times 14 = \frac{2}{3}x - \frac{2}{3} \times 7$$
3. **Calculate the products:**
$$\frac{4}{7}x + \frac{56}{7} = \frac{2}{3}x - \frac{14}{3}$$
Simplify the fractions:
$$\frac{4}{7}x + 8 = \frac{2}{3}x - \frac{14}{3}$$
4. **Eliminate fractions by multiplying both sides by the least common denominator (LCD):**
The LCD of 7 and 3 is 21.
Multiply both sides by 21:
$$21 \times \left(\frac{4}{7}x + 8\right) = 21 \times \left(\frac{2}{3}x - \frac{14}{3}\right)$$
5. **Distribute 21:**
$$21 \times \frac{4}{7}x + 21 \times 8 = 21 \times \frac{2}{3}x - 21 \times \frac{14}{3}$$
Simplify each term:
$$3 \times 4x + 168 = 7 \times 2x - 7 \times 14$$
$$12x + 168 = 14x - 98$$
6. **Isolate variable terms on one side and constants on the other:**
Subtract $12x$ from both sides:
$$\cancel{12x} + 168 = 14x - 12x - 98$$
$$168 = 2x - 98$$
Add 98 to both sides:
$$168 + 98 = 2x - 98 + 98$$
$$266 = 2x$$
7. **Solve for $x$ by dividing both sides by 2:**
$$\frac{266}{\cancel{2}} = \frac{2x}{\cancel{2}}$$
$$133 = x$$
**Final answer:** $$x = 133$$