Problem 46 - Algebra 1

Question

Simplify the expression. $$3 \cdot\left(-\frac{y}{3}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Constant and the Fraction

In the expression \(3 \cdot\left(-\frac{y}{3}\right)\), 3 is our constant, and \(-\frac{y}{3}\) is our fraction.

Step 2: Apply the Rule of Multiplying a Constant with a Fraction

When applying the multiplication rule, the constant is multiplied with the numerator of the fraction and the denominator remains the same. So the expression becomes: \(3 \cdot (-y)\) divided by \(3\).

Step 3: Simplify Further

Multiplying 3 and \(-y\), we get \(-3y\), and now, the fraction becomes: \(-3y/3\). When you divide \(-3y\) by 3, you get \(-y\).

Key Concepts

Multiplication of Fractions

When dealing with the multiplication of fractions, it’s important to understand each part of the fraction format. A fraction has a numerator (top part) and a denominator (bottom part). To multiply a fraction by a constant or another fraction, you follow these simple rules:

• Multiply the constant with the numerator of the fraction.
• The denominator stays the same unless you are multiplying two fractions, then both the numerators and denominators are multiplied.

For example, consider the expression \(3 \cdot \left(-\frac{y}{3}\right)\). Here, the constant 3 is multiplied with \(-y\) (the numerator), while the denominator remains 3. The expression becomes \(-3y/3\), and can simplify further. Breaking it down like this makes it much easier to handle.

Distributive Property

The distributive property is a useful algebraic tool that helps in simplifying expressions by distributing multiplication over addition or subtraction. In the general form, it can be expressed as: \[ a(b + c) = ab + ac \] This property also aids in simplifying expressions when dealing with fractions. By using the distributive property, you can simplify complex expressions and make calculations more straightforward. In our given expression, it helps visualize how the constant multiplies each part inside parentheses: \(3 \cdot \left(-\frac{y}{3}\right)\) shows that the constant is directly multiplying only one term here, making it simpler to handle.

Constants and Variables

Understanding constants and variables is fundamental in algebra. Constants are fixed numbers that don't change, whereas variables are symbols that represent unknown values and can be manipulated. In the expression \(3 \cdot \left(-\frac{y}{3}\right)\):

• The number 3 is a constant, as it’s a fixed value.
• The letter \(y\) is a variable, indicating it can vary based on the context.

By recognizing these elements, simplifying the expression becomes more intuitive. As you perform operations, keep constants distinct from variables to accurately simplify and solve expressions. Knowing how these components interact is essential for tackling more complicated equations in algebra.