Problem 62 - Algebra 1: Concepts and Skills

Question

SIMPLIFYING EXPRESSIONS Simplify the expression. (Lesson 2.7) $$ 3(y-10)-5 y $$

Step-by-Step Solution

Answer

The simplified expression is \(-2y - 30\).

Step 1: Distribution

Start by distributing the 3 across the equation within the parentheses: 3*y - 3*10. Which gives \(3y - 30\).

Step 2: Combine Like Terms

Next, subtract 5y from the result obtained in step 1. Combine like terms: \(3y -5y - 30\). Which gives \(-2y - 30\).

Key Concepts

Distribution Property

One of the foundational skills in algebra is understanding how to distribute a number across the terms inside parentheses. In our exercise, we start with the expression 3(y-10). To simplify, we need to multiply the number outside the parentheses, which is 3, with each term inside the parentheses. This is the distribution property at work. What this essentially means is we are 'sharing' the 3 with the y and -10, giving us 3*y and 3*-10, or in mathematical terms, 3y - 30.

This property is crucial because it allows us to eliminate the parentheses and prepare the expression for further simplification, such as combining like terms, which is our next step. Always remember that every term inside the parentheses must be multiplied by the number outside for correct distribution.

Combine Like Terms

Once we've distributed correctly, the next move is to 'combine like terms'. But what does that mean? Like terms are terms that have the same variable raised to the same power. In our example, after distribution, we have the terms 3y and -5y. Both have the variable y to the power of 1, even though it's not explicitly written (since anything to the power of 1 is itself). Combining them is just basic arithmetic; we subtract 5y from 3y, yielding -2y. It is essential to keep track of positive and negative signs when combining like terms, as they will affect the result.

Remember, only like terms can be combined — so variables must match, and exponents must match. In our example, there are no other like terms, so the -30 remains unchanged in the expression.

Algebraic Operations

Algebraic operations, such as addition, subtraction, multiplication, and division, govern how we manipulate algebraic expressions. During the simplification process, these operations must be applied following specific rules. In the exercise, we do not have any addition or division, but we are using subtraction and multiplication.

The distribution step used multiplication. When we combined like terms, we used subtraction. Grasping these operations in algebra is critical because they are the tools that help us simplify any expression, no matter how complicated it might initially look. In algebra, always perform operations within parentheses first, followed by multiplication and division from left to right, and finally, addition and subtraction from left to right.

Expression Simplification

The ultimate goal of applying the distribution property and combining like terms is to simplify algebraic expressions to their simplest form. Simplifying makes them easier to understand and work with, especially in more complex equations and functions. In our problem, after applying the distribution property and combining like terms, we arrived at -2y - 30. This is the expression in its simplest form — it cannot be simplified any further because there are no more like terms to combine and no further algebraic operations that can be applied.

Simplification is a critical step in solving equations because it often reveals the structure of the problem more clearly and makes the path to the solution more straightforward. Always double-check your work to ensure all like terms are combined and that all operations are correctly applied for a neat and simplified final expression.