Problem 61 - Algebra 1

Question

COMBINING LIKE TERMS Apply the distributive property. Then simplify by combining like terms. $$ (3 y+1)(-2)+y $$

Step-by-Step Solution

Answer

The simplified expression is \(-5y -2\).

Step 1: Applying the Distributive Property

Begin by applying the distributive property to the equation. This means multiplying each term within the parentheses by -2. So you will have \((-2) \times 3y + (-2) \times 1\). That simplifies to \(-6y -2\).

Step 2: Combining Like Terms

Now, add the third term which was outside the parentheses, the y. So \(-6y -2 + y\) which simplifies to \(-5y -2\).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic rule that allows us to multiply a single term by each term within a parenthesis. This property is essential for simplifying complex expressions and is used in the first step of our example exercise.

As seen in the exercise, \(3 y+1)(-2) + y\), we are given a binomial \(3y + 1\) multiplied by \-2\. The distributive property directs us to multiply \-2\ by each term inside the parentheses separately: \

\((-2) \times 3y = -6y)\)

and \( (-2) \times 1 = -2\).

\Putting these results together, we obtain the simplified expression \-6y - 2\. This step expands the expression, setting the stage for further simplification through combining like terms.

Understanding the distributive property is crucial because it's not only used in algebra but also in applied mathematics, including finance, engineering, and physical sciences. Once mastered, it becomes a powerful tool for simplifying and solving equations.

Simplify Expressions

Simplifying expressions is the process of making algebraic expressions easier to work with. After distributing, as shown in the previous section, we often get expressions with multiple terms that can be combined to reduce complexity. This process involves identifying and merging like terms — terms that have the exact variable raised to the same power.

In our exercise, once we have applied the distributive property and expanded the expression to \-6y - 2\, we notice there's another term \(y\) that can be integrated. Like terms can be combined by adding or subtracting their coefficients. Here, we have \-6y\ and \(y\), which are like terms because they both contain the variable \(y\) raised to the first power. By combining these terms, we significantly simplify the expression:

\\

\( (-6y) + (y) = -5y\).

\So, the expression now becomes \-5y - 2\, which is much cleaner and more manageable. Simplifying an expression is advantageous for solving equations since it reduces the potential for errors and makes the logic underlying the solution much clearer.

Algebraic Operations

Algebraic operations are the set of procedures used to manipulate algebraic expressions and solve equations. These include addition, subtraction, multiplication, division, and factoring, among others. When applying these operations, like terms should be combined, and algebraic properties, such as distributive, associative, and commutative properties, should be used effectively.

In the context of our exercise, we first used the distributive property (an algebraic operation) to expand the expression. Next, we performed the operation of addition to combine the like terms. Algebraic operations are often used in sequences, as one step may set up the equation for the next operation. Mastering these operations enables students to approach a variety of problems with the competence necessary to simplify and eventually solve them. Remember, practice is key. Regular exercises involving algebraic operations will sharpen your skills and boost your numerical fluency.

Furthermore, recognizing which operation to apply and when is an essential skill, and being able to foresee the results of an operation before performing it can save time and reduce errors during problem-solving.