Problem 6 - Elementary Algebra

Question

For exercises 1-10, (a) solve. (b) check. $$ \frac{1}{6} w+\frac{23}{8}=-3 $$

Step-by-Step Solution

Answer

w = -35.25

Step 1 - Isolate the variable term

Subtract \(\frac{23}{8}\) from both sides of the equation to isolate the term containing \(w\): \( \frac{1}{6} w = -3 - \frac{23}{8} \)

Step 2 - Simplify the equation

Combine the fractions on the right-hand side of the equation: \( -3 = -3 \times \frac{8}{8} = \frac{-24}{8} \). Now the equation becomes: \[-3 - \frac{23}{8} = \frac{-24}{8} - \frac{23}{8} = \frac{-47}{8} \]So we have \(\frac{1}{6} w = -\frac{47}{8} \)

Step 3 - Solve for the variable

Multiply both sides of the equation by \( 6 \) to solve for \( w \): \[ w = \frac{-47}{8} \times 6 \Rightarrow w = - \frac{282}{8} \Rightarrow w = - \frac{141}{4} \Rightarrow w = - 35.25 \]

Step 4 - Check the solution

Substitute \( w = -35.25 \) back into the original equation and confirm both sides are equal:\[\frac{1}{6} \times (-35.25) + \frac{23}{8} = -3\]Simplify the left-hand side:\[ - \frac{35.25}{6} + \frac{23}{8} = - \frac{5.875}{1} + \frac{2.875}{1} = - 3\]Since both sides are equal, the solution is confirmed.

Key Concepts

Fraction Operations

When working with linear equations, fraction operations play a critical role. A fraction operation involves basic arithmetic—addition, subtraction, multiplication, and division—with fractions.
In this problem, we encounter fractions like \(\frac{1}{6}\) and \(\frac{23}{8}\). Let's break down the fraction operations performed in the given solution:

• Subtraction: We need to isolate the variable term \( \frac{1}{6} w \) by subtracting \(\frac{23}{8}\) from both sides: \[ -3 - \frac{23}{8} = \frac{-24}{8} - \frac{23}{8} = \frac{-47}{8} \]

Notice how we converted \-3\ from a whole number to a fraction \(-3 = -3 \times \frac{8}{8} = \frac{-24}{8}\) to perform the subtraction more easily.
• Multiplication: To isolate \( w \), we multiply both sides by the reciprocal of \( \frac{1}{6} \), which is 6: \[ w = \frac{-47}{8} \times 6 \]

This simplifies to \[ w = \frac{-282}{8} = - 35.25 \]

Understanding these fraction operations ensures you can confidently manipulate expressions involving fractions. It also lays the foundation for solving linear equations effectively.

Variable Isolation

Variable isolation is the process of rearranging an equation to get the variable by itself on one side. This is a fundamental step in solving linear equations.
Here’s how we isolate the variable in the given solution:

• Identify the Variable Term:
  First, we find the variable term in the equation, which is \(\frac{1}{6} w \).


• Isolate the Variable Term:
  To isolate the term, subtract the constant term \( \frac{23}{8} \) from both sides:
  \