Problem 95 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(77-96,\) simplify each algebraic expression. $$4(2 y-3)-(7 y+2)$$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression \(4(2y - 3) - (7y + 2)\) is \(y - 14\).

Step 1: Distribute Multiplication

Using the distributive property of multiplication, distribute the \(4\) to both terms inside the first parenthesis \((2y - 3)\), and the \(-1\) (implied before the second set of parentheses) to both terms inside the second parenthesis \((7y + 2)\):\n\n\(4 * 2y - 4 * 3 - 7y - 2\)

Step 2: Simplify Multiplication Results

To simplify the calculations, perform the multiplication from the previous step:\n\n\(8y - 12 - 7y - 2.\)

Step 3: Combine Like Terms

Lastly, combine like terms to further simplify the expression:\n\nCombine \(8y\) and \(-7y\) to get \(y\), and combine \(-12\) and \(-2\) to get \(-14\):\n\n\(y - 14\).

Key Concepts

Distributive Property

When dealing with expressions involved in multiplication and addition, the distributive property is a very useful technique to understand and apply. The distributive property states that a multiplication distributed over an addition or subtraction means you multiply each term inside the parenthesis by the term outside. For instance, if you have an expression like \(a(b + c)\), you can rewrite it as \(ab + ac\). This property is especially handy when simplifying expressions and is often one of the first steps in solving algebraic problems.

In our specific exercise, we started with \(4(2y - 3)\), and applied the distributive property by multiplying \(4\) with both \(2y\) and \(-3\). This resulted in \(8y - 12\).
Similarly, the expression \(-(7y + 2)\) can be seen as \(-1(7y + 2)\), which allows us to distribute \(-1\) through the terms inside the parentheses, resulting in \(-7y - 2\).

By understanding and applying the distributive property, you can break down and simplify complex expressions efficiently.

Combining Like Terms

Once you've used the distributive property to clear out parentheses, the next step in simplifying an expression is to combine like terms. Like terms are terms that have the same variable raised to the same power. For instance, in the expression \(8y - 12 - 7y - 2\), the terms \(8y\) and \(-7y\) are considered like terms because they both involve the variable \(y\).

To combine like terms, add or subtract their coefficients. In our case, \(8y - 7y\) simplifies to \(y\). Similarly, the constant terms \(-12\) and \(-2\) combine to give \(-14\).

This step greatly reduces the complexity of the expression. It turns multiple terms into significantly fewer, more manageable terms. Understanding how to identify and combine like terms is crucial for algebraic manipulation and simplification.

Algebraic Simplification

Algebraic simplification is a process that aims to reduce expressions to their simplest form, making them easier to understand and work with. It involves using various algebraic techniques to rewrite expressions in a more concise form.

The simplification process generally involves performing operations like:

• Using the distributive property to eliminate parentheses.
• Combining like terms to reduce the number of terms.

In our example, which began as \(4(2y-3)-(7y+2)\), simplification resulted in a straightforward expression without parentheses: \(y - 14\).

The goal of simplification is often to make solving equations or performing further operations much easier. It's a critical skill in algebra that, once mastered, makes tackling even complex problems a much simpler task.