Problem 63 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{x}{2}=6 $$

Step-by-Step Solution

Answer

The solution is \(x = 12\).

Step 1: Understand the Equation

We are given the equation \(\frac{x}{2} = 6\). This is a simple linear equation where the variable \(x\) is divided by 2.

Step 2: Eliminate the Fraction

To solve for \(x\), we need to eliminate the fraction. Multiply both sides of the equation by 2 to cancel out the denominator: \(2 \times \frac{x}{2} = 2 \times 6\).

Step 3: Simplify the Equation

Simplifying the left side of the equation, the 2s cancel out, leaving us with \(x = 12\).

Step 4: Verify the Solution

Substitute \(x = 12\) back into the original equation to verify: \(\frac{12}{2} = 6\), which is a true statement. The solution is verified.

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra that involves finding the values of variables that make an equation true. When dealing with an equation like \(\frac{x}{2} = 6\), our goal is to isolate the variable \(x\).
In its simplest form, this means performing operations on both sides of the equation. The key is to maintain the balance, or equivalently, the equality between the two sides.

• Identify the variable and understand its relationship within the equation.
• Apply inverse operations. Since \(x\) is divided by 2, multiplying by 2 will cancel out the division.
• Perform the operation on both sides: \(2 \times \frac{x}{2} = 2 \times 6\), simplifying to \(x = 12\).

This process of isolating the variable provides a solution for the equation; here, \(x = 12\).

Fractions in Equations

Fractions often appear in linear equations and require specific operations to handle them efficiently. With the equation \(\frac{x}{2} = 6\), the fraction involves dividing \(x\) by 2. To eliminate the fraction:

• Multiply both sides by the denominator of the fraction. Here, that means multiplying by 2.
• The operation affects both sides equally, maintaining the equation's balance. In this case, it simplifies the left side to \(x\).

Removing fractions helps to simplify the equation, making it easier to solve by converting it into a format without fractions. The goal is always to isolate the variable, making it straightforward to find its value.

Verification of Solutions

Verification is a crucial step after finding a solution to an equation. It ensures that the solution is correct and satisfies the original equation. For this problem:

• Substitute the found value of \(x\) back into the original equation \(\frac{x}{2} = 6\).
• Using \(x = 12\), we substitute to check: \(\frac{12}{2} = 6\).
• If the computation holds true, as it does here, the solution \(x = 12\) is verified.

Verification provides confidence in the accuracy of your result, ensuring the integrity of your solution process. It is a valuable practice for any mathematical problem involving equations.