91Ó°ÊÓ

When solving equations, it's important to verify that your solution is correct. By checking solutions, you ensure that the values you found satisfy the original equation. In this exercise, after finding the solution \( r = -13 \), we substitute it back into the original equation \( \frac{r-3}{8} = -2 \).

Here's how it's done:
- Substitute \( r = -13 \) into the expression to get \( \frac{-13-3}{8} \).
- Simplify the expression inside the fraction to \( \frac{-16}{8} \).
- Simplify further to get \( -2 \).Since both sides of the equation equal \( -2 \), the solution checks out. Always remember: Checking your solution is a crucial step, not only to confirm accuracy but to deepen your understanding of the equation.

One of the fundamental steps in solving equations is isolating the variable you are solving for. By isolation of variables, we mean manipulating the equation until the variable is on one side of the equation by itself. This allows us to clearly see what value the variable equals to.

Here’s how we isolated \( r \) in the equation:
- Initially, we had the equation \( \frac{r-3}{8} = -2 \).
- Our first step was to eliminate the fraction by multiplying both sides of the equation by 8, resulting in \( r - 3 = -16 \).
- The next step involved adding 3 to both sides, so \( r - 3 + 3 = -16 + 3 \).
- This simplified to \( r = -13 \).Isolating the variable can involve various arithmetic operations, such as addition, subtraction, multiplication, or division, depending on the equation's structure and the form to be achieved.

Arithmetic operations involve the basic mathematical operations: addition, subtraction, multiplication, and division. These operations are used to manipulate equations and help in transforming them to simplify or solve them.

In the provided exercise, arithmetic operations played a key role:
- **Multiplication:** To eliminate the denominator and simplify the fraction \( \frac{r-3}{8} \) into a simpler form, we multiplied the entire equation by 8.
- **Addition:** After multiplying, we had \( r - 3 = -16 \). To find \( r \), we added 3 to both sides of the equation. This is a straightforward method to move constants around and isolate the variable.These operations are essential because they provide clear pathways to transforming equations from complex forms to simpler equations that are easier to handle and solve.