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Chapter 2: Problem 34

Solve the given equation for \(\mathrm{x}\). \(\frac{x}{2}=11\)

Short Answer

Expert verified
The solution to the equation is \( x = 22 \).

Step by step solution

01

Understand the equation

The given equation is \( \frac{x}{2} = 11 \). This is a simple division equation where \( x \) is divided by 2 resulting in 11.
02

Isolate the variable \(x\)

To find \( x \), we need to get \( x \) alone on one side of the equation. Since \( x \) is currently divided by 2, we can do the opposite operation to both sides of the equation to isolate it: multiply both sides by 2.
03

Multiply both sides by 2

Multiply both sides of the equation by 2 to cancel out the division: \[ \frac{x}{2} \times 2 = 11 \times 2 \]. This simplifies to \( x = 22 \).
04

Verify the solution

To ensure the solution is correct, substitute \( x = 22 \) back into the original equation: \( \frac{22}{2} = 11 \). Since this is a true statement, the solution is verified.

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Key Concepts

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

Division Equations
Division equations are equations that include division as an operation. Often, they will look like \( \frac{x}{a} = b \). Here, the variable \( x \) is divided by a number, turning it into a simpler expression. Division equations are not as daunting as they might seem at first glance.

Understanding how to solve them is a vital skill in algebra:
  • Division in a mathematical equation can seem complex, but it's essentially about distributing a number into equal parts.
  • To solve a division equation, you'll often need to "undo" the division, which involves multiplying.
  • This is because multiplication is the opposite operation of division.

In our example, the division equation is \( \frac{x}{2} = 11 \).

Isolation of Variables
Isolation of the variable is a method where we try to get the unknown, typically \( x \), on one side of the equation entirely by itself. This helps us find the value of the unknown, which solves the equation.

The process is straightforward:
  • Look at both sides of the equation and determine what needs to be done to isolate the variable. For division equations, performing the inverse operation is key.
  • In the equation \( \frac{x}{2} = 11 \), \( x \) is divided by 2. To isolate \( x \), multiply both sides by 2.
  • This gives you \( x = 22 \), effectively isolating \( x \) on one side.

By doing the same operation on both sides, we maintain the balance of the equation.
Verification of Solutions
Verification of solutions ensures the answer is indeed correct. This reinforces understanding and minimizes the possibility of errors in calculations.

Here's how verification is done:
  • Take the solution you found for \( x \), in this case, \( 22 \), and substitute it back into the original equation.
  • Replace \( x \) in the equation \( \frac{x}{2} = 11 \) with 22, resulting in \( \frac{22}{2} \).
  • Simplify the expression to see if both sides of the equation remain equal: \( \frac{22}{2} = 11 \), which verifies that the solution is correct.

This final step confirms the solution, giving confidence in the method and answer used.

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