Problem 96 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(77-96,\) simplify each algebraic expression. $$5(3 y-1)-(14 y-2)$$

Step-by-Step Solution

Answer

The simplified expression is \(y - 3\).

Step 1: Distribute the Coefficients

Multiply the number outside the brackets by each term inside the brackets. For the first bracket, distribute 5 to each term. So, \(5(3y - 1)\) becomes \(15y - 5\). For the second bracket, distribute -1 (since there's a minus sign before the bracket, it's the same as a subtraction), so \(-(14y - 2)\) becomes \(-14y + 2\). So, our expression becomes \(15y - 5 - 14y + 2\).

Step 2: Combine Like Terms

Combine terms with the same variables. So, the term \(15y\) and \(-14y\) are like terms and can be combined. Also, terms \(-5\) and \(+2\) are like terms. So, our expression simplifies to \(1y - 3\).

Step 3: Final Simplified Expression

The expression simplifies to \(y - 3\), since \(1y\) is the same as \(y\).

Key Concepts

Distributive Property

Understanding the distributive property is key to simplifying algebraic expressions. It's a tool that allows us to multiply a single term by each of the terms inside a parenthesis. For instance, in the expression 5(3y-1), we apply the distributive property by multiplying 5 times 3y and 5 times -1, yielding 15y - 5. Similarly, when we face a subtraction, such as -(14y - 2), it's helpful to consider the negative sign as a -1 coefficient, resulting in -14y + 2 after distribution. This step sets the stage for further simplification by setting up the expression for the next important concept: combining like terms.

Keep in mind that distributing correctly avoids common mistakes and it's essential for dealing with more complex expressions which involve multiple terms and variables. Always ensure each term within the parentheses is multiplied by the term outside.

Combining Like Terms

Once the distributive property has been applied, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our example, once we distributed the coefficients, we obtained 15y - 5 - 14y + 2. The like terms here are 15y and -14y, both terms of 'y'. To combine them, simply add their coefficients to get 15 - 14, which equals 1, resulting in 1y or simply y, since the coefficient 1 is implicit.

Similarly, the constants -5 and +2 are like terms too, combining to yield -3. It's essential to maintain accuracy during this process as mistakes in combining like terms can lead to incorrect answers. Be attentive to the signs of each term and perform the necessary addition or subtraction to combine them correctly.

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. Expressions represent quantities and do not have an equality sign unlike equations. The algebraic expression from our example, 5(3y-1)-(14y-2), went through transformation steps to reach its simplest form, y - 3. Understanding how to work with and simplify expressions is foundational in algebra.

The beauty of algebraic expressions lies in their ability to model real-world situations and their flexibility to simplify or transform to reveal properties or facilitate easier calculations. Learning to identify parts of an expression and apply simplification rules like distribution and combining like terms will be invaluable skills as you progress in mathematics.