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Chapter 8: Problem 99

Solve. $$(x-5) 1 / 2-2=0$$

Short Answer

Expert verified
The solution is \(x = 9\).

Step by step solution

01

Eliminate Fractions

Multiply both sides of the equation by 2 to eliminate the fraction: \((x - 5) \cdot \frac{1}{2} - 2 = 0\). This simplifies to \((x - 5) - 4 = 0\).
02

Isolate the Variable Term

Add 4 to both sides of the equation to isolate the term with \(x\): \((x - 5) = 4\).
03

Solve for x

Add 5 to both sides of the equation to solve for \(x\): \(x = 4 + 5\).
04

Simplify the Solution

Simplify to find \(x = 9\).

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Key Concepts

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

Eliminating Fractions
To solve equations effectively, it's often helpful to eliminate any fractions involved. Fractions can complicate calculations and obscure the steps needed to isolate a variable. The most straightforward way to get rid of a fraction is by multiplying every term in the equation by the denominator of the fraction.
For example, in the equation \((x - 5) \cdot \frac{1}{2} - 2 = 0\), the fraction \(\frac{1}{2}\) is present. To eliminate this fraction:
  • Multiply each term by 2, the denominator of the fraction.
  • This operation will cancel the fraction, rewriting the equation as \((x - 5) - 4 = 0\).
After this step, the equation becomes much simpler, paving the way to easily isolate the variable and solve the equation.
Isolating Variable
Once you have eliminated any fractions, the next goal is to isolate the variable in the equation. Isolating a variable means getting the variable by itself on one side of the equation. This is important because it allows you to identify the solution to the equation directly.
In our example, we proceed from \((x - 5) - 4 = 0\) to isolate \(x\):
  • Add 4 to both sides of the equation to cancel out the \(-4\) term, simplifying the equation to \((x - 5) = 4\).
  • Now, \(x\) is not completely isolated yet, but this step clears out any other constants interacting directly with \(x\).
Each operation, such as adding or subtracting, must be performed on both sides to maintain the equation's balance. Isolating the variable gets us one step closer to finding its value.
Simplifying Equations
Simplifying an equation is about making it as straightforward as possible, often involving a few more steps after isolating the variable. Step-by-step simplification makes it easier to see the underlying relationships within the equation and reach the solution more smoothly.
Continuing with our equation \((x - 5) = 4\):
  • Add 5 to both sides to finally isolate \(x\): this updates the equation to \(x = 4 + 5\).
  • The expression \(4 + 5\) naturally simplifies to \(9\).
At each step, ensure you perform the same operation on both sides to keep the equation balanced and valid. Simplifying confirms the exact value of the variable, giving a clear and concise answer.

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