Problem 38 - Elementary Algebra

Question

Simplify. $$ 1 y-1+11-y $$

Step-by-Step Solution

Answer

The simplified expression is 10.

Step 1: Combine like terms

Look at all the terms in the expression you have: \( 1y - 1 + 11 - y \). Notice that you have two terms with \( y \). These are \( 1y \) and \( -y \). They can be combined together.

Step 2: Simplify the y terms

Combine the \( y \) terms: \( 1y - y = 0y \). Since \( 0y = 0 \), the resulting expression is just the constants.

Step 3: Simplify the constant terms

Now focus on the number terms: \( -1 + 11 \). When you add these constants, you get \( -1 + 11 = 10 \).

Step 4: Write the simplified expression

After combining like terms, the simplified expression becomes just the constant \( 10 \).

Key Concepts

Understanding Like Terms

When working with algebraic expressions, a key step is recognizing and working with _like terms_. Like terms are terms in an expression that have the same variables raised to the same powers. For instance, in the expression \( 1y - 1 + 11 - y \), the terms \( 1y \) and \( -y \) are like terms because they both contain the variable \( y \).

Understanding like terms is important because it allows you to combine them and simplify expressions. Here are some tips to identify like terms:

• Check if they have identical variable parts.
• Ensure that these variables are raised to the same powers.

Identifying and working with like terms is crucial for simplifying algebraic expressions, and it will allow you to focus on the components that can be combined.

Combining Like Terms

Once you've identified the like terms in an expression, the next step is to combine them. This process involves adding or subtracting their coefficients—the numbers in front of the variable part.

Let's revisit our expression \( 1y - 1 + 11 - y \). We recognize the like terms here: \( 1y \) and \( -y \). To combine these, you adjust their coefficients:

• \( 1y - y = 0y \)

Since \( 0y = 0 \), you've effectively removed the variable part from your expression.

Combining like terms simplifies your expression and makes it easier to work with. It's a helpful skill in algebra that reduces complexity by collapsing terms that can naturally belong together.

Working with Constant Terms

Now that we've dealt with like terms, we can turn our attention to the constant terms. Constant terms are numbers without variables. They stand alone and don't change based on the values of variables.

In the expression \( 1y - 1 + 11 - y \), the constant terms are \( -1 \) and \( 11 \). Combining constant terms is straightforward because it involves simple arithmetic:

• Add or subtract the constants: \( -1 + 11 = 10 \).

After you've combined them, the expression is left as just \( 10 \), which is your fully simplified form.

Dealing with constant terms is a critical step in simplifying algebraic expressions, allowing you to reach a numeric answer cleanly and clearly.