Problem 66 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ 3(y-4)-5(y+3) $$

Step-by-Step Solution

Answer

-2y - 27

Step 1: Distribute

To begin solving the expression, apply the distributive property to both terms inside the parentheses. This involves multiplying the constant outside the parentheses by each term inside: \[ 3(y - 4) - 5(y + 3) = 3 imes y - 3 imes 4 - 5 imes y - 5 imes 3 \] Simplifying this: \[ = 3y - 12 - 5y - 15 \]

Step 2: Combine Like Terms

Now, combine the like terms in the expression. The like terms here are the ones containing \(y\) and the constant terms: Combine \(3y\) and \(-5y\): \[ 3y - 5y = -2y \] Combine the constants \(-12\) and \(-15\): \[ -12 - 15 = -27 \] So now the expression is: \[ -2y - 27 \]

Key Concepts

Combining Like Terms

When working with algebraic expressions, it is essential to know how to "combine like terms." This means you group and simplify terms in an expression that share the same variable or variables to the same power. For the exercise with the expression \(3(y-4)-5(y+3)\), after distributing, you get the terms \(3y\), \(-5y\), \(-12\), and \(-15\).

In this context, "like terms" are terms that have the same variable raised to the same power. Here:

• \(3y\) and \(-5y\) both contain the variable \(y\).
• \(-12\) and \(-15\) are constant numbers.

Combining like terms means you add or subtract the numerical coefficients of the terms with variables and separate the constants. In this exercise:

• Combine \(3y - 5y\) which equals \(-2y\).
• Combine \(-12 - 15\) which equals \(-27\).

Thus, by combining like terms, the expression simplifies to \(-2y - 27\). This operation helps condense the expression, making it simpler and easier to interpret or use in further calculations.

Polynomial Expressions

Polynomial expressions are mathematical expressions made up of variables and constants using addition, subtraction, and multiplication. In the context of algebra, a polynomial can have one or more terms. Each term in a polynomial is usually made up of a coefficient (a number) and a variable raised to an exponent.In the exercise \(3(y-4)-5(y+3)\), after distributing and expanding: \[3y - 12 - 5y - 15\], we recognize this as a polynomial expression. It contains four separate terms:

• \(3y\)
• \(-5y\)
• \(-12\)
• \(-15\)

Within polynomial expressions, the degree is determined by the highest power of the variable. Here, since each term involving \(y\) is raised to the first power, the highest degree of the polynomial is 1, making it a linear expression.By recognizing and understanding the elements of polynomial expressions, students can better evaluate and simplify them in algebraic equations. It facilitates the process of simplifying expressions or carrying out operations such as addition, subtraction, and factoring.

Algebraic Simplification

Algebraic simplification is the process of reducing an algebraic expression to its simplest form. This involves a combination of different algebra operations, including applying the distributive property, combining like terms, and reducing fractions where applicable. The goal is to make an expression as easy and clear as possible for further calculation or evaluation.Using the exercise \(3(y-4)-5(y+3)\), we can see algebraic simplification at work. After distributing the terms \(3(y-4)\) and \(-5(y+3)\):

• \(3y - 12\)
• \(-5y - 15\)

The next step is to combine like terms to streamline the expression:

• \(3y - 5y\)
• \(-12 - 15\)

This process results in the simpler expression \(-2y - 27\).Algebraic simplification helps unveil the underlying structure of an expression, making equations easier to solve or understand. By stripping away unnecessary complexity, it allows for a further focus on the essential parts of an algebraic expression.