Problem 80 - Introductory Algebra for College Students

Question

Simplify each algebraic expression. $$-5\left(-\frac{3}{5} y\right)$$

Step-by-Step Solution

Answer

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

Step 1: Distribute the multiplication

To simplify the given expression, distribute -5 into the term insides parentheses. Specifically, -5 should be multiplied by \(-\frac{3}{5}y\). Using the multiplication rule for fractions, this results in \[-5 \times - \frac{3}{5} y = 3y.\]

Step 2: Result

After simplifying the expression, the result is \(3y\). This is achieved due to the fact that multiplying two negative values yields a positive result. Therefore, \(-5 \times -\frac{3}{5}y\) simplifies to \(3y\).

Key Concepts

Simplification

Simplification in algebra involves breaking down expressions into their simplest form. This means reducing the expression so it cannot be simplified further. For the expression \(-5\left(-\frac{3}{5} y\right)\), simplification involves applying operations correctly to turn the expression into a cleaner and more manageable form like \(3y\).

• Identify the terms within the expression: In this case, the expression inside the parentheses is \(-\frac{3}{5} y\).
• Focus on simplifying step-by-step: Multiply each term accurately by distributing operations.

Taking time to master simplification can help solve many algebra problems efficiently.

Multiplication Rule for Fractions

The multiplication rule for fractions is essential for handling algebraic expressions involving fractions. When multiplying a number by a fraction, you multiply the number by the numerator and keep the denominator unchanged.

When multiplying \(-5\) by \(-\frac{3}{5} y\), perform the following steps:

• Multiply \(-5\) by \(-3\), the numerator of the fraction: \((-5) \times (-3) = 15\).
• Because you are multiplying a whole number by a fraction, the denominator remains 5.
• Thus, the fraction simplifies to \(\frac{15}{5} y\).
• Finally, \(\frac{15}{5} = 3\), giving the simplified result of \(3y\).

Understanding this rule can make working with fractions in algebra much simpler.

Negative Numbers in Multiplication

When dealing with negative numbers in multiplication, remember that multiplying two negative numbers results in a positive number. This rule is key to correctly simplifying expressions like \(-5\left(-\frac{3}{5}y\right)\).

• Negative \(-5\) and negative \(-\frac{3}{5}\) multiply to form a positive product: \((-5) \times (-3) = 15\).
• Since both numbers are negative, their product becomes positive, turning negative into positive.
• Understanding this concept is crucial in algebra to avoid sign errors that can lead to incorrect answers.

Grasping the multiplication of negative numbers ensures that students handle expressions with confidence.