you say multiply both sides, why isn’t x-5 being multiplied by x+7?

Great question! When I said "multiply both sides by \( x - 5 \)," I was referring to eliminating the fraction by multiplying each side of the equation by the denominator \( x - 5 \). 

Let's clarify that step:

### The Original Equation:
\[
2 = \frac{x + 7}{x - 5}
\]

### Goal:
To eliminate the fraction, multiply **both sides** of the equation by the denominator \( x - 5 \).

### Step-by-Step Breakdown:

1. **Multiply the left side** by \( x - 5 \):
   \[
   2 \times (x - 5)
   \]
   
2. **Multiply the right side** by \( x - 5 \):
   \[
   \frac{x + 7}{x - 5} \times (x - 5)
   \]
   
   Here, the \( x - 5 \) in the numerator and denominator cancel each other out, leaving:
   \[
   x + 7
   \]

### Result After Multiplication:
\[
2(x - 5) = x + 7
\]

### Why Doesn’t \( x - 5 \) Multiply \( x + 7 \)?
The \( x - 5 \) cancels out with the denominator on the right side when multiplied, leaving just \( x + 7 \). On the left side, \( x - 5 \) is multiplied by 2, giving \( 2(x - 5) \).

This is why you don't see \( x - 5 \) multiplying \( x + 7 \); it's because the multiplication is effectively "undoing" the fraction, simplifying the equation.

Would you like more details, or do you have any other questions? Here are some related questions for further practice:

1. Why do we need to eliminate fractions when solving equations?
2. What happens if the denominator contains a variable that makes the denominator zero?
3. Solve \( \frac{x + 4}{x - 3} = 3 \).
4. Simplify the expression \( \frac{(x + 7)(x - 2)}{(x - 5)(x + 3)} \).
5. How can you check if a solution to a rational equation is valid?

**Tip:** Always be cautious about the values that make the denominator zero, as these are not valid solutions.

Explore the concept of multiplying equations to eliminate fractions. Learn how to apply this method with detailed examples and step-by-step explanations. Suitable for high school students and those preparing for algebraic equations.

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