Problem 58 - Algebra 1: Concepts and Skills

Question

Solve the equation. Round the result to the nearest hundredth. Check the rounded solution. $$14-6 r=-17$$

Step-by-Step Solution

Answer

The solution to the equation is \( r = 5.17 \).

Step 1: Isolate the term with \( r \)

In the equation \( 14-6 r=-17 \), first move \( 14 \) to the right side by subtracting \( 14 \) from both sides. We get \( -6r = -17 - 14 \) which simplifies to \( -6r = -31 \).

Step 2: Solve for \( r \)

To solve for \( r \), divide both sides by \( -6 \) to get \( r = -31 / -6 \). This gives \( r = 5.166666667 \).

Step 3: Round to nearest hundredth

The value of \( r \) needs to be rounded to the nearest hundredth, which gives \( r = 5.17 \).

Step 4: Check the solution

To check the rounded solution, substitute \( r = 5.17 \) into the original equation: \( 14 - 6*5.17 = -17 \), which evaluates to approximately \( -17 \). Since the right-hand side of the equation is \( -17 \), the solution is correct.

Key Concepts

Isolating Variables

In mathematics, solving an equation often begins with the process of isolating the variable you are trying to solve for. This means getting the variable all by itself on one side of the equation. Consider the equation from our problem:
\[ 14-6r=-17 \].

• First, identify the term containing the variable. In this case, it is \(-6r\).
• To isolate \(r\), the first step is to remove the constant term \(14\) from the left side of the equation. This can be done by subtracting \(14\) from both sides, ensuring you keep the equation balanced. This results in:
  \[-6r = -17 - 14\].
• Now, the next task is to deal with the coefficient of \(r\), which is \(-6\). To isolate \(r\), divide every term by \(-6\):
  \[r = \frac{-31}{-6}\].

Through these steps, you can see that isolating a variable refers to systematically performing operations that help separate the variable from other numbers or terms.

Rounding Numbers

Rounding is an essential skill in mathematics that simplifies numbers while still keeping them close to their original values. In our equation problem, once we found the solution for the variable \(r\) as \[ r = 5.166666667 \],we needed to round it to the nearest hundredth.
To round to the nearest hundredth:

• Identify the digit in the hundredths place. For our solution, the hundredths digit is \(6\).
• Look at the digit to the right (the thousandths place) to decide whether to round up or down. This digit is also \(6\). Since it's 5 or greater, increase the hundredths digit by one.
• The rounded value for \(r\) becomes
  \[ r = 5.17 \].

Rounding not only makes calculations easier but is often necessary when dealing with real-world measurements and constraints where extreme precision is unnecessary.

Checking Solutions

Once you've solved for a variable and possibly rounded your results, it’s crucial to check your solution to ensure accuracy. Here’s how we verify if our answer \( r = 5.17 \) is indeed correct:

• Start with the original equation:
  \[ 14 - 6r = -17 \].
• Substitute the rounded solution for \(r\):
  \[ 14 - 6 \times 5.17 = -17 \].
• Perform the multiplication:
  \[ 6 \times 5.17 = 31.02 \].
• Subtract this product from \(14\):
  \[ 14 - 31.02 = -17.02 \].
• The result is approximately
  \[-17\], which matches the right side of the equation closely.

The small difference arises due to rounding, but the solution is considered correct since it is within an acceptable range of precision. This process ensures not only that your answer is correct but also builds confidence in solving similar problems.