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Chapter 2: Problem 14

For exercises 11-46, (a) solve. (b) check. $$ d-\frac{1}{8}=\frac{23}{8} $$

Short Answer

Expert verified
\(d = 3\)

Step by step solution

01

- Isolate the variable

To solve for the variable d, add \( \frac{1}{8} \) to both sides of the equation to get rid of the fraction on the left side.\[ d - \frac{1}{8} + \frac{1}{8} = \frac{23}{8} + \frac{1}{8} \] Simplifies to \[ d = \frac{24}{8} \]
02

- Simplify

Simplify \( \frac{24}{8} \) to get the value of d.\[ d = 3 \]
03

- Check the solution

Plug the value of d back into the original equation to verify.\[ 3 - \frac{1}{8} = \frac{23}{8} \]Convert 3 to a fraction with the same denominator: \[ \frac{24}{8} - \frac{1}{8} = \frac{23}{8} \]As both sides of the equation are equal, 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 the linear equation for the variable d, our first goal is to isolate the variable on one side of the equation. Start with the equation:
  1. Subtract or add fractions: Handling fractions can be tricky. Always check to align them correctly. For this equation, we add \frac{1}{8}\ to both sides to move the term away from d. Applying the operation, we get: \[ d - \frac{1}{8} + \frac{1}{8} = \frac{23}{8} + \frac{1}{8} \]
  2. Simplify: After performing the calculation, we simplify the equation. The left side of the equation now just has the variable d: \[ d = \frac{24}{8} \]. By isolating d, we can easily determine its value in the next step.
Simplifying fractions
In our previous step, we isolated the variable and reached the fraction \( \frac{24}{8} \). Now, let's simplify. Simplifying fractions makes it easier to understand and use the result in further calculations.

  1. Identify a common divisor: Both 24 and 8 can be divided by 8.
  2. Divide the numerator and the denominator by 8: \[ \frac{24 \div 8}{8 \div 8} = \frac{3}{1} \], which simplifies to \( 3 \).
    So, the value of d is now determined to be 3.
Check solutions
Finally, let's check our solution. It's important to verify that our answer satisfies the original equation.

  1. Substitute d with our computed value: Replace d in the original equation with 3: \[ 3 - \frac{1}{8} = \frac{23}{8} \]
  2. Convert whole numbers to fractions: Convert 3 to a fraction with denominator 8, which gives us \( \frac{24}{8} \). Now our equation looks like: \[ \frac{24}{8} - \frac{1}{8} = \frac{23}{8} \].
  3. Simplify: Perform the calculation on the left side: \[ \frac{24-1}{8} = \frac{23}{8} \]. Both sides are equal, confirming our solution is correct.
    Remember to always substitute back and check if both sides of the equation match. This ensures the solution is accurate and consistent.

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Most popular questions from this chapter

For exercises 89-92, solve. Use a calculator to do the arithmetic. $$ -\frac{3}{8}=\frac{b}{160} $$

Solve \(A=\frac{x_{1}+x_{2}+x_{3}}{3}\) for \(x_{3}\).

Solve: \(7(2 c-1)=15(c-3)-c\)

a. Solve \(A=P+P R T\) for \(R\). b. An investment of \(\$ 500\) grows to a future value of \(\$ 650\) in 5 years. Find the simple interest rate.

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