Problem 23 - Elementary Algebra

Question

Solve each of the equations. $$\frac{x-1}{2}-1=\frac{3}{4}$$

Step-by-Step Solution

Answer

The solution is \( x = \frac{9}{2} \).

Step 1: Clear the Fraction by Multiplying

The equation is \( \frac{x-1}{2} - 1 = \frac{3}{4} \). To eliminate fractions, find the least common multiple of denominators, which is 4, and multiply every term by 4: \[ 4 * \left(\frac{x-1}{2}\right) - 4 * 1 = 4 * \frac{3}{4} \] This simplifies to: \[ 2(x-1) - 4 = 3 \]

Step 2: Distribute and Simplify

Distribute the 2 in the equation: \[ 2(x-1) = 2x - 2 \] Substitute back into the equation: \[ 2x - 2 - 4 = 3 \] Simplify by combining like terms: \[ 2x - 6 = 3 \]

Step 3: Solve for x

Add 6 to both sides of the equation to isolate terms involving \( x \): \[ 2x = 3 + 6 \] \[ 2x = 9 \] Divide both sides by 2 to solve for \( x \): \[ x = \frac{9}{2} \] Thus, the solution is \( x = \frac{9}{2} \).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is an essential skill in solving equations. It involves re-arranging and simplifying expressions to find the value of unknown variables. In this exercise, algebraic manipulation plays a key role in each step.
First, we need to simplify expressions and move all terms to one side of the equation. By combining like terms or eliminating constants, the equation becomes simpler. This makes it easier to isolate the variable we want to solve for.
For example, in the given exercise, after multiplying to clear fractions, we distribute and simplify terms such as distributing the 2 over the \(x - 1\) expression. This helps us transition the equation from \(2(x-1) - 4 = 3\) to \(2x - 6 = 3\).
Following simplification, adding or subtracting numbers across the equation sets you up to isolate the variable—known as rearranging the equation. Wrapping up with division or multiplication allows algebraic manipulation to yield the final value.

Fractions in Equations

Fractions often appear in equations and can sometimes make solving them seem difficult. However, there's a systematic way to handle them. The key is to eliminate the fractions early on to simplify the process.
To begin with, determine the least common multiple (LCM) of all denominators in the equation. Multiply each term on both sides of the equation by this LCM. This approach converts the fractions into whole numbers, making the algebraic manipulation more straightforward.
In this exercise, multiplying all terms by 4 removes the fractions, changing \(\frac{x-1}{2}-1=\frac{3}{4}\) to \(2(x-1) - 4 = 3\). From this point, it becomes easier to manipulate the equation.
This method is a critical problem-solving technique, especially when fractions hinder simple algebraic processing.

Step-by-Step Solutions

Solving algebraic equations using a step-by-step approach is beneficial for clear understanding. This methodical technique involves breaking down the problem into simple, manageable pieces.
Here’s a streamlined path followed:

• Identifying the need to clear fractions by using the least common multiple of the denominators.
• Executing algebraic manipulation through distribution and simplification of verbal expressions.
• Solving for the variable by isolating it and simplifying all accompanying terms.

By addressing each step sequentially, we ensure nothing is overlooked and every stage is comprehended. Solving \(\frac{x-1}{2} - 1 = \frac{3}{4}\) begins with eliminating fractions, distributing terms, and isolating the variable, concluding with solving \(x = \frac{9}{2}\).
This systematic step-by-step approach provides a foolproof way to master algebraic equation solving.