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Chapter 8: Problem 132

Solve the equation for the variable. $$3 y+\frac{1}{2}=2 y$$

Short Answer

Expert verified
y = -\frac{1}{2}

Step by step solution

01

Move all terms with the variable to one side

Subtract 2y from both sides of the equation to get the variable terms on one side: \[ 3y - 2y + \frac{1}{2} = 2y - 2y \] This simplifies to: \[ y + \frac{1}{2} = 0 \]
02

Isolate the variable

Subtract \( \frac{1}{2} \) from both sides of the equation to isolate y: \[ y + \frac{1}{2} - \frac{1}{2} = 0 - \frac{1}{2} \] This simplifies to: \[ y = -\frac{1}{2} \]

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Key Concepts

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

Isolating Variables
Isolating variables is a critical skill in algebra that allows you to solve for an unknown. In this context, it means getting the variable 'y' by itself on one side of the equation. Let's consider the original problem: \[ 3y + \frac{1}{2} = 2y \].
The first step is to move all terms containing 'y' to one side. This is important because it allows you to combine like terms and simplify the equation.
We subtract '2y' from both sides: \[ 3y - 2y + \frac{1}{2} = 2y - 2y \] Simplifying this, we get: \[ y + \frac{1}{2} = 0 \] Now 'y' is almost by itself. The next task is to isolate 'y' completely.
Simplifying Equations
Simplifying equations makes them easier to solve. After isolating the variable 'y', you may still have terms that need simplifying. For instance, from the prior step, we have: \[ y + \frac{1}{2} = 0 \] To simplify, we need to eliminate the fraction standing in the way of isolating 'y'. We do this by performing the same operation on both sides. In this case, we subtract \( \frac{1}{2} \) from both sides: \[ y + \frac{1}{2} - \frac{1}{2} = 0 - \frac{1}{2} \] This leaves us with: \[ y = - \frac{1}{2} \] This simplification step is crucial for solving equations efficiently. Always remember to balance your equation by doing the same operation on both sides.
Algebra Steps
Mastering algebra involves understanding and applying a series of logical steps. Let's recap the steps for the exercise we worked on:
  • Identify the terms with the variable you need to solve for.
  • Move all terms with the variable to one side by adding or subtracting.
  • Simplify the equation by combining like terms and performing arithmetic operations equally on both sides.
  • Isolate the variable by performing the inverse operation.

In our case, we followed these steps to get from: \[ 3y + \frac{1}{2} = 2y \] to: \[ y = - \frac{1}{2} \]
Following each step carefully ensures you can handle similar equations with confidence. Take your time to understand each step, and you'll find solving linear equations becomes much easier.

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