Problem 64 - Intermediate Algebra

Question

Simplify by combining like terms. See Example 5 . $$12 y-17 y$$

Step-by-Step Solution

Answer

The simplified expression is \(-5y\).

Step 1: Identify Like Terms

The expression is given as \(12y - 17y\). Here, the terms are \(12y\) and \(-17y\), which are like terms because they both contain the variable \(y\) and can be combined.

Step 2: Combine Like Terms

To combine \(12y\) and \(-17y\), keep the variable \(y\) and perform the subtraction: \(12 - 17 = -5\). Thus, the combined expression is \(-5y\).

Key Concepts

Combining Like Terms

When you hear the term "combining like terms," think of organizing and simplifying an expression by putting similar items together. A "like term" is simply a term that has the same variables raised to the same power. For example, in the expression \(12y - 17y\), both terms are like terms because they contain the variable \(y\) to the same power of 1.

Here's a simple way to look at it:

• Identify like terms: Same variables, same exponents.
• Combine them by calculating the coefficients (numbers in front of the variables).

In summary, when combining like terms, you are basically performing addition or subtraction on their coefficients, while keeping the variable part unchanged. This helps to reduce the expression to its simplest form.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations. Variables are symbols (like \(x\) or \(y\)) that represent unknown values and can vary. These expressions are foundational in algebra since they form the building blocks for more complex problem-solving.

In the expression \(12y - 17y\), "12" and "-17" are coefficients that are multiplied by the variable \(y\). Expressions can become simpler or more manageable when we correctly combine like terms, which is a critical step in solving equations or inequalities.

Remember:

• An algebraic expression doesn’t include an equality sign—it’s not an equation until it does.
• Knowing how to identify terms in an expression is key to any simplification.

Understanding and manipulating these expressions are essential skills for progressing in algebraic problem-solving.

Intermediate Algebra

Intermediate algebra focuses on building and refining skills learned in basic algebra. It includes working with more complicated expressions and performing operations that prepare students for topics in calculus and beyond.

When dealing with tasks like combining like terms in Expressions such as \(12y - 17y\), you are honing your ability to simplify expressions, which is critical in intermediate algebra.

Some key concepts in intermediate algebra include:

• Solving equations and inequalities.
• Understanding and applying functions.
• Simplifying complex algebraic fractions and expressions.

These skills are crucial for tackling more advanced mathematical ideas and will serve you well as you delve deeper into the world of mathematics.