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Chapter 5: Problem 323

In the following exercises, solve the equation. $$ \frac{r}{-1.2}=-6 $$

Short Answer

Expert verified
r = 7.2

Step by step solution

01

- Understand the Equation

The equation given is \ \( \frac{r}{-1.2} = -6 \). It is a simple algebraic equation where \( r \) is the variable that needs to be solved.
02

- Isolate the Variable

To isolate \( r \), multiply both sides of the equation by -1.2, the denominator of the fraction. This can be written as: \ \( \frac{r}{-1.2} \times (-1.2) = -6 \times (-1.2) \)
03

- Perform the Multiplication

Simplify both sides by performing the multiplication. On the left side, the \( -1.2 \) cancels out: \ \( r = -6 \times (-1.2) \). On the right side, calculate \( -6 \times -1.2 \): \ \( r = 7.2 \)
04

- Verify the Solution

Substitute \( r = 7.2 \) back into the original equation to ensure it is correct: \ \( \frac{7.2}{-1.2} = -6 \). Simplifying this, we get: \ \( -6 = -6 \), confirming the solution is correct.

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Key Concepts

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

isolate the variable
To solve algebraic equations, the first crucial step is to isolate the variable you want to find. In our example, the variable is \( r \) in the equation \( \frac{r}{-1.2} = -6 \). To isolate \( r \), we need to eliminate the fraction. You can do this by multiplying both sides of the equation by the denominator, which is -1.2. This action ensures that the value of \( r \) stands alone on one side of the equation. So the operation looks like this:
\( \frac{r}{-1.2} \times (-1.2) = -6 \times (-1.2) \).
Notice how \( -1.2 \) on the left side cancels out, leaving us with \( r \). This is the essence of isolating the variable - making sure it鈥檚 by itself on one side of the equation.
simplify both sides
Now that we have isolated our variable \( r \), the next step is to simplify both sides of the equation. Starting with what was left of the previous step, we have:
\( r = -6 \times -1.2 \).
When you perform the multiplication, remember the rule about multiplying negative numbers: a negative times a negative is a positive. In this case: \( r = 7.2 \).
This simplification step is essential. Always carefully perform arithmetic operations to avoid mistakes. This will ensure you get the correct value for the variable.
verify the solution
The final step in solving any algebra problem is to verify the solution. This means plugging the value you found back into the original equation to see if it holds true.
Substitute \( r = 7.2 \) into the original equation: \( \frac{7.2}{-1.2} = -6 \).
When you simplify the left side, \( \frac{7.2}{-1.2} \), you should get \( -6 \), which matches the right side of the equation.
So, \( -6 = -6 \).
This confirms that our solution is correct. Verification helps catch any potential errors made during the problem-solving process.

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