Problem 10 - Algebra 2

Question

Solve each equation. Check each solution. $$ \frac{1}{4}-x=\frac{x}{8} $$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{1}{4} - x = \frac{x}{8}\) is \(x = \frac{2}{9}\)

Step 1: Eliminate fractions

Multiply every term by 8 (the LCM of 4 and 8) to eliminate fractions. \n \( 8 * \frac{1}{4} - 8x = 8 * \frac{x}{8} \) which simplifies to \n \( 2 - 8x = x \)

Step 2: Simplify

Move the \(x\)-terms to one side of the equation and constant terms to the other. \n \( 2 = 8x + x \) which simplifies to \n \( 2 = 9x \)

Step 3: Solve for \(x\)

Divide both sides by 9 to solve for \(x\). \n \( \frac{2}{9} = x \)

Step 4: Check the solution

Substitute value of \(x\) back into the original equation to verify the solution. \n \( \frac{1}{4} - \frac{2}{9} =? \frac{2}{9*8} \) LHS (left hand side) simplifies to \n \( \frac{9 - 8}{36} = \frac{1}{36} \) and RHS (right hand side) simplifies to \n \( \frac{1}{36} \) Since LHS = RHS, the solution is correct.

Key Concepts

Understanding Linear Equations

A linear equation is an algebraic expression where each term is either a constant or the product of a constant and a single variable. It usually takes the form \(ax + b = c\). The ultimate goal when working with linear equations is to isolate the variable \(x\) or other designated unknown.

• Start by simplifying both sides of the equation if necessary.
• Combine like terms to make the equation less complex.
• Make logical operations such as adding, subtracting, multiplying, or dividing both sides of the equation by the same number to maintain equality.

By following these steps, you can systematically solve most linear equations. Understanding the importance of maintaining balance across both sides of the equation is key to finding the accurate value for the variable.

Handling Fractions in Equations

Equations with fractions might seem intimidating at first, but they can be analyzed similarly to other linear equations. The previous examples show us how to tackle fractions:

Focusing on eliminating fractions can often simplify the process significantly. A common method is the Multiplication of all terms by the Least Common Multiple (LCM) of the denominators:

• This gets rid of the fractional components, leaving you with an equation involving only whole numbers.
• In our example, we multiplied every term by 8, resulting in \(2 - 8x = x\).

Simplifying the equation in this manner helps maintain clarity and efficiency, avoiding potential complications that arise from handling fractional terms individually.

The Importance of Checking Solutions

After solving an equation, it’s crucial to verify that the solution is correct. This step not only ensures the accuracy of the question but reinforces your understanding of the solution process and its application back to the original problem.

• Substitute the found value of \(x\) back into the original equation to verify both sides are equal.
• Check that both the LHS (Left Hand Side) and RHS (Right Hand Side) simplify to the same result.

Our equation confirmed the solution was correct because substituting \(x = \frac{2}{9}\) returned balanced sides of the equation: both reduced to \(\frac{1}{36}\). This validation step not only fosters confidence in your solution but also enhances mastery over checking and solving techniques.