Problem 79 - Elementary and Intermediate Algebra

Question

Simplify by combining like terms. $$ 15 y-10-y-20 y $$

Step-by-Step Solution

Answer

The simplified expression is \(-6y - 10\).

Step 1: Identify Like Terms

In the expression \(15y - 10 - y - 20y\), identify the terms that can be combined. The like terms here are the ones with the \(y\) variable: \(15y\), \(-y\) (which is the same as \(-1y\)), and \(-20y\). The constant term \(-10\) does not have any like terms.

Step 2: Combine the Y Terms

Add and subtract the coefficients of the \(y\) terms: \(15y - y - 20y\). Simplify this by calculating: \(15 - 1 - 20 = -6\). So the combination of the \(y\) terms will be \(-6y\).

Step 3: Combine All Terms

Now that you've combined the \(y\) terms to get \(-6y\), the expression simplifies to \(-6y - 10\). There are no more like terms, so this is your simplified expression.

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental concept in algebra that involves making equations or expressions easier to work with. In algebra, expressions can look complex, but they can be simplified by following a series of steps, including combining like terms.

• First, identify terms in the expression that can be combined. These include terms with the same variable raised to the same power.
• Next, perform arithmetic operations like addition and subtraction to combine these terms.
• Simplification opens the door to easier manipulation of algebraic equations, making them more straightforward to solve or analyze.

Breaking down the process of simplification enhances understanding and aids in solving more complicated algebra problems effectively.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. Understanding algebraic expressions is vital in learning algebra, as they are foundational elements used in equations and inequalities.

• Variables, like the letter 'y' in the expression, stand in for unknown values and make algebraic expressions versatile.
• Operations such as addition, subtraction, multiplication, and division show how terms within the expression interact with each other.
• Coefficients, like 15 or -1, are the numerical part of terms that multiply the variable.

Learning to work with algebraic expressions involves not only recognizing their components but also understanding the relationships and rules that govern them.

Like Terms in Algebra

Recognizing and combining like terms is crucial for simplifying algebraic expressions. Like terms refer to terms that have the same variables raised to the same powers. This allows them to be easily grouped and simplified through basic arithmetic operations.

• In the exercise, terms such as \(15y\), \(-y\), and \(-20y\) are like terms because they all contain the variable \(y\).
• Unlike terms, such as \(-10\), do not have a similar variable and cannot be combined with terms that do.
• When simplifying, ensure correct combination of like terms by appropriately adding or subtracting their coefficients.

Mastering the identification and combination of like terms helps streamline the simplification of expressions and aids in solving algebraic equations more easily.