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Chapter 2: Problem 48

Solve equation, and check your solution. \(\frac{r}{8}=-3\)

Short Answer

Expert verified
The solution is \(r = -24\).

Step by step solution

01

Identify the Equation

The given equation is \ \ \ \ \ \ \ \(\frac{r}{8} = -3\).
02

Isolate the Variable

To isolate the variable \(r\), multiply both sides of the equation by 8. \ \ \ \ \(\frac{r}{8} \cdot 8 = -3 \cdot 8\).
03

Perform the Multiplication

Solve the equation from Step 2. \ \ \ \ \(r = -24\).
04

Check the Solution

Substitute \(r = -24\) back into the original equation to verify. \ \ \ \ \(\frac{-24}{8} = -3\). \ \ \ \ Since \(-3 = -3\), the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolation of Variables
Isolation of variables is an essential part of solving algebraic equations. In this problem, the goal is to isolate the variable r in the equation \(\frac{r}{8} = -3\). To do this, we need to eliminate the fraction. We achieve this by multiplying both sides of the equation by 8鈥攖he denominator of the fraction with r.

This step aims to get r alone on one side of the equation, turning our fraction into a whole number. Multiplying both sides by 8, we get:

\(\frac{r}{8} \cdot 8 = -3 \cdot 8\)

After simplification, this becomes:

\(r = -24\)

By isolating r, we find its value clearly and can proceed to the next steps more efficiently.
Basic Algebra
Basic algebra involves understanding and manipulating equations to find the values of unknown variables. In our example, we start with a simple linear equation: \(\frac{r}{8} = -3\). To break this down:
  • The equation tells us that r divided by 8 equals -3.
  • Our task is to find the value of r that makes this equation true.


In basic algebra, we use operations like addition, subtraction, multiplication, and division to transform the equation and isolate the variable, as demonstrated here. Multiplying both sides by 8, the equation \(\frac{r}{8} \cdot 8 = -3 \cdot 8\) simplifies to \(r = -24\). Understanding these fundamental operations is key to mastering algebra and solving more complex equations in the future.
Checking Solutions
After finding the value of the variable, it's crucial to verify the solution to ensure it satisfies the original equation. In our example, we've found that \(r = -24\). Now, let's check if this solution is correct by substituting r back into the original equation \(\frac{r}{8} = -3\)

Substituting \(r = -24\):

\(\frac{-24}{8} = -3\)

We simplify the left side:

\(-3 = -3\)

Since both sides of the equation equal -3, the solution \(r = -24\) is correct. This step is essential to ensure accuracy and confirm that our algebraic manipulations were performed correctly.
It eliminates any mistakes we might have made during the process, giving us confidence in our final answer.

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