Problem 80 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(77-96,\) simplify each algebraic expression. $$-5\left(-\frac{3}{5} y\right)$$

Step-by-Step Solution

Answer

The simplified form of the given algebraic expression \(-5\left(-\frac{3}{5} y\right)\) is \(3y\).

Step 1: Identify the expression

The given expression is \(-5\left(-\frac{3}{5} y\right)\).

Step 2: Apply the concept of multiplication

Multiplication is distributive over subtraction. Hence, multiply \(-5\) with \(-\frac{3}{5} y\) which gives \(3y\).

Step 3: Simplification

After simplifying, the final expression is \(3y\) which is the simplified form of the given expression.

Key Concepts

Understanding Multiplication in Algebra

Multiplication in algebra involves combining numbers and variables to create new expressions. When multiplying a number by a variable, you distribute the number to each term inside the expression. In the given problem, it’s important to multiply

• the coefficient, which is \(-5\),
• with the expression inside the parentheses, \(-\frac{3}{5} y\).

Remember, when multiplying fractions, multiply the numerators together and the denominators together. For example, multiplying \(-5\) by \(-\frac{3}{5} y\) involves multiplying the numbers \(-5\) and \(-3/5\). This results in a positive product because multiplying two negative numbers results in a positive number.

Exploring the Distributive Property

The distributive property is a useful algebraic tool that allows you to simplify expressions and solve equations efficiently.
It states that a term outside a set of parentheses can be distributed to each term inside the parentheses. In mathematical terms, for any numbers \(a\), \(b\), and \(c\), the distributive property is expressed as:

• \(a(b + c) = ab + ac\).

In this exercise, the expression \(-5\left(-\frac{3}{5} y\right)\) uses the distributive property, where \(-5\) is multiplied directly with the term inside the parentheses, \(-\frac{3}{5} y\).
Using this property correctly ensures simplification results in the final solution, which here is \(3y\).
The property is particularly helpful when dealing with binomials or polynomials, but even in simpler cases, it ensures accuracy.

Handling Negative Numbers

Dealing with negative numbers can be challenging, but it’s crucial for mastering algebra.
Negative numbers are those less than zero, and they behave differently than positive numbers, especially in multiplication. Here are key rules:

• Multiplying two negative numbers results in a positive product.
• Multiplying a positive and a negative number results in a negative product.

In the expression \(-5\left(-\frac{3}{5} y\right)\), both numbers involved are negative. Thus, their product is positive, leading to the simplified term \(3y\).
Understanding these rules helps simplify complex expressions effortlessly and accurately. Negative numbers can initially seem intimidating, but with practice, they become a straightforward aspect of algebra.