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Chapter 2: Problem 48

Solve using the multiplication principle. Don't forget to check! $$ -68=-r $$

Short Answer

Expert verified
The solution is \( r = 68 \).

Step by step solution

01

- Isolate the variable r

To isolate the variable, multiply both sides of the equation by -1. This will help us to find the positive value of \( r \).
02

- Apply the multiplication principle

Multiply both sides of the equation \( -68 = -r \) by -1. This gives \( -1 \times (-68) = -1 \times (-r) \). Simplifying, we get \( 68 = r \).
03

- Check the solution

Substitute \( r = 68 \) back into the original equation to verify. The original equation is \( -68 = -r \). Substitute \( r \) to get \( -68 = -68 \), which confirms that \( r = 68 \) is correct.

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Key Concepts

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

solving equations
Solving equations is a fundamental skill in algebra. It involves finding the value of the variable that makes the equation true. In this problem, we start with the equation \( -68 = -r \). The goal is to determine the value of \( r \) that satisfies this equation. The process typically involves manipulation steps to simplify or rearrange the equation. Common techniques include addition, subtraction, multiplication, and division to isolate the variable. These operations are often referred to as maintaining the balance of the equation.
variable isolation
Variable isolation means getting the variable alone on one side of the equation so we can clearly see its value. In our problem, \( -68 = -r \), we want to isolate \( r \). The step involves multiplying both sides by \( -1 \). When we do this, we change the sign of both sides, so:

\[ -1 \times (-68) = -1 \times (-r) \]

This equals \( 68 = r \), successfully isolating \( r \). Here, we鈥檝e used the multiplication principle, which allows us to maintain the equality by performing the same operation on both sides.
checking solutions
After isolating the variable, it's important to check if our solution is correct. To check the solution, substitute the value back into the original equation and see if it holds true. For our solution, \( r = 68 \), substitute back into the original equation \( -68 = -r \). Replacing \( r \) with \( 68 \), we get:

\[ -68 = -(68) \]

Simplifying this, we have \( -68 = -68 \), which confirms that our solution is correct. This step ensures that no computational errors were made, and the solution reliably satisfies the original equation.

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