Problem 26 - Algebra A Combined Function

Question

Solve each equation. See Examples 3 through \(5 .\) $$ \frac{5}{2} x-1=x+\frac{1}{4} $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{5}{6}\).

Step 1: Eliminate Fractions by Finding a Common Denominator

To eliminate fractions, we multiply every term by the common denominator of the fractions present. Here, the denominators are 2 and 4. The least common multiple is 4. So, multiply every term by 4 to get:\[ 4 \left( \frac{5}{2} x \right) - 4(1) = 4(x) + 4 \left( \frac{1}{4} \right) \] which simplifies to:\[ 10x - 4 = 4x + 1 \].

Step 2: Simplify the Equation by Moving Terms

Next, we move all the terms involving \(x\) to one side and constant terms to the other side. Subtract \(4x\) from both sides:\[ 10x - 4x - 4 = 4x - 4x + 1 \]This simplifies to:\[ 6x - 4 = 1 \].

Step 3: Isolate the Variable Term

To isolate \(6x\), add 4 to both sides of the equation:\[ 6x - 4 + 4 = 1 + 4 \]This simplifies to:\[ 6x = 5 \].

Step 4: Solve for x

Finally, solve for \(x\) by dividing both sides by 6:\[ \frac{6x}{6} = \frac{5}{6} \]So, \(x = \frac{5}{6} \).

Key Concepts

Common Denominator

When dealing with equations that include fractions, finding a common denominator can simplify the process. This concept stems from comparing fractions with different denominators to make them easier to manipulate within an equation. In our example, the fractions have denominators of 2 and 4. We find their least common multiple (LCM), which is 4.

Once you determine the common denominator, you can multiply all terms in the equation by this number. This action eliminates the fractions entirely, transforming the equation into a simpler form where all terms are whole numbers.

Eliminating Fractions

The process of eliminating fractions involves converting an equation from fraction form to a whole number format, which is often easier to work with. This is achieved by multiplying every term in the equation by the common denominator.

In the given exercise, multiplying each term by 4 effectively eliminates the fractions. The expression \( \frac{5}{2}x \) becomes \(10x\), and \(\frac{1}{4}\) simplifies to 1. As a result, we have the linear equation \(10x - 4 = 4x + 1\). This process simplifies the algebraic manipulation and reduces the likelihood of mistakes.

Simplifying Equations

After eliminating fractions, the next step is to simplify the equation. Simplification involves rearranging terms to make the equation as straightforward as possible.

For our equation \(10x - 4 = 4x + 1\), we move terms involving the variable \(x\) to one side and constants to the other. Subtracting \(4x\) from both sides gives \(6x - 4 = 1\), showing a cleaner form that is easier to interpret and solve in the subsequent steps.

Isolating Variable

Isolating the variable is crucial for solving equations, as it allows us to determine its specific value. Once the terms are simplified, we focus on isolating the term involving the variable on one side of the equation.

In this exercise, starting from \(6x - 4 = 1\), we add 4 to both sides, simplifying it to \(6x = 5\). Here, all non-variable terms are removed from the side with \(x\), bringing us closer to solving the equation.

Variables in Equations

Understanding variables and their manipulation is key to solving algebraic equations. A variable, like \(x\) in the equation, is an unknown value we need to find.

The process involves several steps—finding a common denominator, eliminating fractions, simplifying terms, and isolating the variable. Once isolated, further operations such as division can lead to finding the precise value of the variable. Successfully navigating these steps allows us to conclude that in the equation \(\frac{5}{2}x - 1 = x + \frac{1}{4}\), the solution for \(x\) is \(\frac{5}{6}\).

Mastering these concepts will empower you to tackle a wide range of linear equations, laying a strong foundation for more advanced mathematical endeavors.