Problem 6 - Algebra and Trigonometry Real Mathematics, Real People

Question

Fill in the blank. How can you clear the equation \(\frac{x}{2}+1=\frac{1}{4}\) of fractions?

Step-by-Step Solution

Answer

The equation \(\frac{x}{2}+1=\frac{1}{4}\) can be cleared of fractions by multiplying through by the least common denominator, which in this case is 4. This produces the equation \(2x + 4 = 1\), which simplifies to \(2x = -3\) and finally \(x = \frac{-3}{2}\).

Step 1: Multiply every term by the Least Common Denominator (LCD)

In this equation, \(\frac{x}{2}+1=\frac{1}{4}\), the denominators are 2 and 4. The least common denominator (LCD) is 4. Therefore, multiply every term in the equation by 4. You get \(4*\frac{x}{2}+4*1=4*\frac{1}{4}\)

Step 2: Simplify the equation

Simplify the multiplications in the equation. Multiplying \(4*\frac{x}{2}\) gives 2x. For the term \(4*1\), the result is 4. And for the term \(4*\frac{1}{4}\), it simplifies to 1. The equation becomes \(2x+4=1\).

Step 3: Rearrange the equation to isolate x

The equation \(2x+4=1\) needs to be rearranged to isolate x. Therefore, subtract 4 from both sides of the equation to get \(2x=1-4\), which simplifies to \(2x=-3\).

Step 4: Solve for x

Finally, solve for x by dividing both sides of the equation \(2x=-3\) by 2. Therefore, \(x=\frac{-3}{2}\).

Key Concepts

Least Common Denominator

The Least Common Denominator (LCD) is a foundational concept when dealing with fractions in equations. The LCD is the smallest number that all denominators in the equation can divide into without leaving a remainder. It's a bit like finding the best meeting time that suits everyone's schedule.

To determine the LCD, list the multiples of each denominator and find the smallest multiple they have in common. In the exercise \(\frac{x}{2}+1=\frac{1}{4}\), the denominators are 2 and 4. The multiples of 2 are 2, 4, 6, etc., and the multiples of 4 are 4, 8, 12, etc. The smallest common multiple is 4, making it the LCD. Multiplying each term of the equation by the LCD removes the fractions, which simplifies the equation. This process is essential because it transforms a fractional equation into a simpler, integer-based equation.

Finding the LCD Helps:

• Eliminate fractions to simplify equations.
• Facilitate the use of integer operations.
• Prepare the equation for solving.

Simplifying Equations

Once the least common denominator is used to eliminate fractions, the next step is simplifying the equation. Simplification makes the equation easier to manage and sets the stage for finding the solution. Essentially, simplifying means performing the necessary arithmetic operations and combining like terms.

Consider the example \(4*\frac{x}{2}+4*1=4*\frac{1}{4}\). After multiplying by the LCD, we simplify the expression by performing the multiplication. We're left with a linear equation \(2x + 4 = 1\), which is much easier to handle than the original version with fractions. Simplifying also involves rearranging terms, if necessary, to get the variable(s) you're solving for on one side of the equal sign.

Tips for Simplifying Equations:

• Perform all the multiplications and divisions first.
• Combine like terms to reduce the equation.
• Keep the equation balanced by performing the same operation on both sides.

Equation Solving

Finally, the goal of equation solving is to find the value of the unknown variable that makes the equation true. This process involves a series of steps that isolate the variable.

In the adjusted equation \(2x + 4 = 1\), we proceed by isolating \(x\). Subtract 4 from both sides to get \(2x = -3\). Next, divide by 2, since \(2x\) means \('2'\) times \(x\), and we want to find out what \(x\) equals on its own. Solving \(2x = -3\) by dividing both sides by 2 gives us \(x = \frac{-3}{2}\), which is the solution. In short, solving an equation is like discovering the missing piece of a puzzle that makes the whole picture clear.

Key Steps to Solve Equations:

• Isolate the variable by reversing the operations surrounding it.
• Maintain balance by doing the same operation to both sides of the equation.
• Check the solution by substituting it back into the original equation.