Problem 75 - Elementary Algebra

Question

Simplify. $$ x-1+y-1 $$

Step-by-Step Solution

Answer

The expression simplifies to \(x + y - 2\).

Step 1: Identify Like Terms

The expression given is \( x - 1 + y - 1 \). Notice that there are two constant terms: \(-1\) and \(-1\). Also, the terms \(x\) and \(y\) can be considered separately as they are not like terms.

Step 2: Combine Like Terms

Add together the constant terms from the expression \(-1 + (-1) = -2\). So, modify the expression to: \(x + y - 2\).

Step 3: Write Simplified Expression

After combining like terms, the expression simplifies to \(x + y - 2\). You now have the simplified version of the original expression.

Key Concepts

Like Terms

In algebra, **like terms** are terms that have the same variable raised to the same power. For example, in the expression \(2x + 3x + y\), the terms \(2x\) and \(3x\) are like terms because they both contain the variable \(x\) raised to the same power. Each of these terms will usually have a coefficient, the number directly in front of the variable.Identifying like terms is crucial when simplifying expressions because it allows us to combine them. When looking at the expression \(x - 1 + y - 1\), it’s important to first spot that \(x\) and \(y\) are not like terms because they contain different variables. This initial recognition guides the simplification process.

Constant Terms

**Constant terms** in algebra are the numbers on their own, without any attached variables. Essentially, they are terms that do not change, regardless of the value of any variables present. In the expression \(x - 1 + y - 1\), the terms \(-1\) and \(-1\) are constant because they do not contain any variables.Constant terms are easy to spot and play a vital role when simplifying expressions, as they can often be combined to reduce the complexity of the equation. When simplifying, it’s important to add up all constant terms, just as you would regular numbers.

Combining Terms

**Combining terms** is a key operation in algebra, allowing for the simplification of expressions. To do this effectively, like terms and constant terms should be combined separately. This means adding or subtracting their coefficients or numerical values while keeping the variables and powers intact.In the expression \(x - 1 + y - 1\), after identifying that \(x\) and \(y\) have no other like terms, we look at the constant terms, \(-1\) and \(-1\). By combining these, you get \(-2\), simplifying the expression to \(x + y - 2\).This process of combining makes the expression cleaner and easier to manage without changing its value. It’s a fundamental skill that allows algebraic expressions to be as simplified and readable as possible.