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Chapter 3: Problem 7

Decide which operation you would use first to solve the equation. $$\frac{y}{2}=0$$

Short Answer

Expert verified
The value of 'y' is equal to 0. The operation first applied to solve the equation was multiplication.

Step by step solution

01

Identify the operation

To isolate 'y', we need to perform multiplication as the reverse operation of the division presented in the problem.
02

Apply the appropriate operation

Multiply both sides of the equation by 2 to cancel out the division by 2. So, \(2 * \frac{y}{2} = 2 * 0\).
03

Simplify the equation

After applying the multiplication, you simplify the equation as \(y = 0\).

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Key Concepts

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

Isolation of Variables
The heart of solving equations lies in isolating the variable we want to solve for. This means getting the variable alone on one side of the equation. By doing this, we can clearly see its value. In the example,we started with \( \frac{y}{2} = 0 \). To isolate \(y\), we need to remove the fraction operation that connects \(y\) to the number 2.
  • Start by identifying the operation being applied to the variable. Here, we're looking at \(y\) being divided by 2.
  • The ultimate goal is to "undo" this operation, or in other words, cancel out the fraction effect on \(y\).

By systematically identifying and removing operations that cloud the variable, we can find precise solutions for our equations. Isolation is about untangling complexity and focusing on what truly matters鈥攖he variable's value.
Inverse Operations
To isolate a variable in an equation, we often use inverse operations. These are operations that "undo" each other, helping us strip away unwanted parts of the equation. In the equation \(\frac{y}{2} = 0\), the division operation needs to be undone.
Using multiplication, which is the inverse of division, we can achieve this:
  • If \(y\) is divided by 2, multiplication by 2 will bring \(y\) back to its full value.
  • This is why we multiply both sides of the equation by 2: \(2 * \frac{y}{2} = 2 * 0\).

Applying inverse operations step-by-step helps us simplify the equation without changing its balance. It's like carefully peeling away layers until only the core鈥攖he variable鈥攊s visible.
Equation Simplification
After performing inverse operations to isolate the variable, we often need the final step of simplification. Simplification makes equations cleaner and easier to read.In our example, once we multiplied both sides by 2, the equation shifted from \(2 * \frac{y}{2} = 0\) to \(y = 0\).
Here's how simplification works:
  • Once inverse operations are applied, terms may cancel out, leaving the variable alone.
  • Removing these extraneous terms helps us clearly understand the equation's resolution.

The process of equation simplification ensures the final result is in its simplest, most understandable form. This is the satisfaction found in mathematics: reducing the complex to the simple.

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