Problem 76 - Elementary Algebra

Question

Simplify. $$ x-2-y-1 $$

Step-by-Step Solution

Answer

The simplified expression is \( x - y - 3 \).

Step 1: Combine Like Terms

First, identify the like terms in the expression. In this case, the like terms are the constant numbers (-2 and -1). The expression can be rewritten so that these numbers are next to each other: \( x - y - 2 - 1 \).

Step 2: Simplify the Constants

Add the constant numbers together. \( -2 - 1 = -3 \). Now the expression becomes \( x - y - 3 \).

Key Concepts

Like Terms

In algebra, like terms are terms that have the same variables raised to the same powers. Identifying like terms is crucial when simplifying expressions. For example, in the expression \( x - 2 - y - 1 \), terms like \( x \) and \( y \) aren't like terms as they contain different variables. However, the numbers \(-2\) and \(-1\) are constant terms and hence considered like terms.

• Like terms must have exactly the same variable(s).
• They allow you to group and simplify an expression more easily.
• Grouping like terms often simplifies additions and subtractions in expressions.

To simplify, you align like terms together. In our example, this involves rearranging -2 and -1 to be combined, helping streamline the simplification process.

Constant Terms

Constant terms are numbers on their own in an expression, free of any attached variable. They hold a fixed value and play an essential role in arithmetic simplification. For instance, in the expression \( x - 2 - y - 1 \), the numbers \(-2\) and \(-1\) are constant terms because they do not have accompanying variables.

• Constant terms remain unchanged as they don't depend on variables.
• Combining constants is a key step in simplifying expressions.
• After like terms are aligned, constants can be condensed into a single number.

In our problem, we found the constant terms \(-2 - 1\) add up to \(-3\), simplifying the expression from its original state.

Expression Simplification

The process of expression simplification involves two core steps: identifying like terms and constantly reducing them. This not only makes mathematical expressions easier to handle, but it also is crucial for solving algebraic equations.Expression simplification often involves:

• Rearranging expressions to align like terms together.
• Performing addition or subtraction on constant terms.
• Simplifying to the smallest possible form without changing the expression's value.

For example, the expression \( x - 2 - y - 1 \) simplifies to \( x - y - 3 \) after combining the like constant terms. This condensed form is easier to understand or use in further equations.