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Chapter 1: Problem 32

Solve the given equation for \(\mathrm{x}\). $$ \frac{x}{4}=6 $$

Short Answer

Expert verified
The solution is \( x = 24 \).

Step by step solution

01

Understand the Problem

The equation given is \( \frac{x}{4} = 6 \). It states that when \( x \) is divided by 4, the result is 6. We need to find the value of \( x \).
02

Isolate the Variable

To find \( x \), we need to eliminate the 4 in the denominator. Multiply both sides of the equation by 4 to cancel out the division by 4 on the left side.
03

Solve for x

Multiply both sides of the equation by 4: \[4 \times \frac{x}{4} = 4 \times 6\] This simplifies to \( x = 24 \).
04

Verify the Solution

Substitute \( x = 24 \) back into the original equation to check: \[\frac{24}{4} = 6\] Since the left side equals the right side, \( x = 24 \) is the correct solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving One-Step Equations
In prealgebra, solving one-step equations is one of the first algebraic skills that students learn. It's simple yet powerful. A one-step equation is an equation that requires only one operation to find the variable.
For example, in the given problem \( \frac{x}{4} = 6 \), the goal is to solve for \( x \) by performing the reverse operation of what's being done to it. Since \( x \) is divided by 4, we need to multiply both sides of the equation by 4 to reverse this operation.
This approach applies to any one-step equation:
  • Identify the operation being done to the variable.
  • Use the inverse operation to both sides to eliminate this.
  • Keep the equation balanced by performing the operation on both sides.
The solution results directly from this simple manipulation.
Isolation of Variables
The next important concept in solving equations is the isolation of variables. This means rearranging the equation so the unknown variable is alone on one side of the equation.
In our example \( \frac{x}{4} = 6 \), the variable \( x \) needs to be isolated. To do this, multiply both sides by 4. It cancels out the division on the left side:
  • Multiply both sides by 4: \( 4 \times \frac{x}{4} = 4 \times 6 \)
  • This simplifies to: \( x = 24 \)
Notice how the multiplication undoes the division, effectively isolating \( x \). This step is crucial in algebra, allowing us to see the value of the unknown clearly.
Equation Verification
Solving an equation is not complete until you verify the solution. This checks if the solution is correct, ensuring the values satisfy the original equation.
For the equation \( \frac{x}{4} = 6 \), we found \( x = 24 \). To verify, substitute \( x = 24 \) back into the original equation:
  • \( \frac{24}{4} = 6 \)
  • The left side equals the right side: \( 6 = 6 \)
Since both sides of the equation are equal, the solution is verified. Verification minimizes errors and confirms that the solving process was accurate. Always include this step to reinforce understanding and accuracy.

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