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Chapter 2: Problem 25

Solve for the indicated variable. \(y=\frac{1}{2}(x+2)\) for \(x\)

Short Answer

Expert verified
The solution to the equation for \(x\) is \(x = 2y - 2\).

Step by step solution

01

Clear the Fraction

Multiply both sides of the equation by 2 to clear the fraction. This gives 2y = x + 2.
02

Isolate xs

Subtract 2 from both sides of the equation to isolate \(x\) on one side. This gives \(x=2y-2\).
03

Final Simplification

There's no further simplification needed, so \(x = 2y - 2\) is the final solution.

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Key Concepts

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

Solving for a Variable
In algebra, solving for a variable involves isolating one variable on one side of the equation. This is a key concept that allows us to express the variable in terms of other known quantities. In the exercise where you need to solve for \(x\) in the equation \(y=\frac{1}{2}(x+2)\), you begin by rearranging the terms to isolate \(x\).
  • First, you address any fractions to simplify the equation.
  • Then, manipulate the equation by performing operations such as addition or subtraction.
  • Finally, verify your answer by substituting it back into the original equation.
The ultimate goal is to express \(x\) solely in terms of the other variable \(y\), which allows for easy computation values if \(y\) is known.
Fractions in Equations
Fractions can often complicate the process of solving equations. When fractions are present, it is usually helpful to eliminate them early on. This makes the equation easier to work with. In the original exercise, you encounter the fraction \(\frac{1}{2}\). To eliminate this:
  • Multiply every term by the denominator (in this case, 2).
  • This multiplication clears the fraction, leading to a simpler form of the equation without fractions.
By converting the equation \(y = \frac{1}{2}(x+2)\) into \(2y = x + 2\), you've successfully removed the fraction. Employing this technique consistently will make solving algebraic equations with fractions much less intimidating.
Algebraic Manipulation
Algebraic manipulation involves using various operations to either simplify or rearrange an equation. This is crucial when solving almost any algebra problem. Here's how it's applied in our exercise:
  • Once fractions are dealt with, focus on placing the variable on one side of the equation.
  • You may need to perform operations like adding, subtracting, multiplying, or dividing across both sides of the equation to maintain equality.
  • Ensure operations are executed properly to prevent errors in calculation.
In our exercise, after clearing the fractions, we subtract 2 from both sides to isolate \(x\).
This operation transforms the equation \(2y = x + 2\) into \(x = 2y - 2\).
Effective algebraic manipulation is essential for solving equations accurately and efficiently.

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