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Cross-multiplication is a powerful tool used to solve equations involving fractions. When you encounter an equation where a variable is in the denominator, like in the given exercise \( 15 = \frac{r}{-5} \), cross-multiplication allows you to isolate the variable by multiplying each side of the equation by the denominators. This process effectively eliminates the fraction.

Here's a step-by-step cross-multiplication for the given problem:
1. Multiply 15 by -5 (the denominator of the fraction).
2. At the same time, multiply the other side of the equation, which is the fraction \( \frac{r}{-5} \), by -5.
3. This operation will make the denominators on both sides cancel out. As a result, you are left with \( -5 \times 15 = r \).

It's important to remember that whatever operation you do to one side of the equation, you must also do to the other side to maintain balance.

Equation simplification is key to finding an easily understandable solution. Once cross-multiplication has been done, the next step involves simplifying the equation to find the value of the variable. Simplification reduces the equation to its simplest form, thus making it easier to solve.

In our example, the cross-multiplication resulted in the equation \( -5 \times 15 = r \). We continue by simplifying the multiplication on the left side:
1. Multiply -5 by 15 to get -75.
2. Now, the equation looks like this: \( -75 = r \).

You have successfully isolated and identified the variable \( r \). It's now expressed as a single number, \( r = -75 \), which represents the solution to the original problem. Simplification is not just about performing operations; it includes checking for common factors, reducing fractions, and ultimately making the problem more straightforward.

After finding the solution to an equation, it's essential to verify that it's correct. This helps you ensure that no mistake was made during the solving process and that the solution satisfies the original equation. Verification is the final step and a good habit for all levels of math practice.

To verify the solution for our equation \( r = -75 \), insert this value back into the original equation:
1. Replace \( r \) with -75 in \( 15 = \frac{r}{-5} \).
2. The equation becomes \( 15 = \frac{-75}{-5} \).
3. Now simplify the right side by dividing -75 by -5 to get 15.
4. The equation now reads \( 15 = 15 \), which shows that both sides are equal.

The solution is verified because the left side is equal to the right side when the proposed solution is used. This step acts as your guarantee that the solution found is indeed appropriate for the equation at hand.