Problem 35 - Elementary Algebra

Question

Evaluate. \(y_{2}-y-6,\) where \(y=0\)

Step-by-Step Solution

Answer

The evaluated result is -6.

Step 1: Identify the Expression

The expression given is \(y_{2} - y - 6\). This means we need to evaluate this expression for the given value of \(y\).

Step 2: Substitute the Given Value

We substitute \(y = 0\) into the expression. This gives us: \(0_{2} - 0 - 6\).

Step 3: Simplify the Expression

Next, we simplify the expression by evaluating each term. Since \(0_{2} = 0\), we have: \(0 - 0 - 6 = -6\).

Key Concepts

Substitution Method

The substitution method is a useful technique in algebra, allowing us to replace variables with their given values to solve or evaluate mathematical expressions or equations. In this exercise, we are given the expression \(y_{2} - y - 6\) and a specific value for \(y\), which is \(0\). By substituting \(y = 0\) into the expression, we replace every instance of the variable \(y\) with the number \(0\). This step is crucial because it transforms an expression with variables into one that only contains numbers, making it simpler to handle.
For clarity, let's see how the substitution unfolds in our example:

• Start with the original expression: \(y_{2} - y - 6\).
• Substitute \(y\) with \(0\): \(0_{2} - 0 - 6\).

This substitution turns an algebraic expression into a numeric one, setting the stage for further simplification.

Simplifying Expressions

Once substitution is done, the next step is simplifying the expression. Simplification involves performing arithmetic operations to reduce the expression to its simplest form. In the given exercise, after substituting \(y = 0\) into the expression \(y_{2} - y - 6\), it becomes \(0_{2} - 0 - 6\).
This step is straightforward:

• Calculate \(0_{2}\). Any number or variable multiplied by zero is zero, so \(0_{2}\) equals \(0\).
• Then you subtract \(0\) and \(6\): \(0 - 0 - 6\).

The simplification process consolidates these into a single number, resulting in \(-6\). Being comfortable with simplifying expressions helps in making subsequent calculations easier and more manageable.

Evaluating Expressions

Evaluating expressions involves finding the value of an algebraic expression by simplifying it after substituting the given values. This concept is fundamental when dealing with algebraic calculations as it brings the computation to a conclusion. After substituting \(y = 0\) and simplifying the expression to \(0 - 0 - 6\), the evaluation process gives us the final result, \(-6\).
Here’s a concise breakdown of the evaluation process:

• Start with the substituted expression: \(0_{2} - 0 - 6\).
• Simplify it to \(0 - 0 - 6\).
• Conclude with the final answer: \(-6\).

Evaluating expressions often involves these essential steps—substitution, simplification, and calculation—to determine the value of the expression, which is especially important for verifying results and solving mathematical problems effectively.