Problem 54 - College Algebra with Modeling and Visualization

Question

Apply the distributive property. $$-6(3 y-2 x)$$

Step-by-Step Solution

Answer

The expression simplifies to \(-18y + 12x\).

Step 1: Identify the Terms Inside the Parentheses

The expression inside the parentheses is \(3y - 2x\). These are the terms that will be multiplied by \(-6\).

Step 2: Distribute the \(-6\) to Each Term

To apply the distributive property, multiply \(-6\) by each term inside the parentheses individually. First, multiply \(-6\) by \(3y\).

Step 3: Multiply -6 by 3y

Calculate \(-6 \times 3y\) which results in \(-18y\).

Step 4: Multiply -6 by -2x

Now multiply \(-6\) by \(-2x\):\(-6 \times -2x = 12x\).

Step 5: Combine the Results

The distributed expression is the sum of the results from Steps 3 and 4: \(-18y + 12x\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that helps simplify expressions. It states that when you multiply a sum by a number, you can multiply each addend by that number separately and then add the products. This concept is useful in making complex mathematical expressions more manageable. When applying it to an expression like

• Starting with the formula: For an expression \( a(b + c) \), distribute \( a \) to both \( b \) and \( c \).
• Apply the formula as: \( a \times b + a \times c \).

This approach prevents the common mistake of forgetting to multiply the number with each term or incorrectly performing multiplication.

Expressions

Expressions in algebra are combinations of numbers, variables, and operations that represent a particular value or set of values. An algebraic expression, unlike an equation, does not contain an equality sign. For instance, the expression \(3y - 2x \) represents a calculation that combines two terms:

• \(3y\), which consists of a coefficient (3) and a variable (\(y\));
• \(-2x\), involving a coefficient (-2) and a variable (\(x\)).

By using expressions, we can model real-world scenarios in a way that can be manipulated and analyzed mathematically. This makes them extremely powerful tools in algebra.

Variables

Variables are symbols used to represent unknown values in expressions and equations. They can stand for a single number or a set of numbers that fit within the problem's context. Typically denoted by letters such as \(x\) and \(y\), variables are placeholders that allow us to perform algebraic operations abstractly.

• Variables can change.
• They help form relationships and functions.
• They simplify the expression manipulation process.

Using variables allows us to generalize mathematical principles, solve equations, and discover patterns.

Multiplication

Multiplication in algebra is one of the core arithmetic operations. It combines quantities so we can understand and work with the relationship between different sets of values. In algebra, multiplication often involves combining numbers and variables:

• When multiplying coefficients with variables, such as \(-6 \times 3y\), we multiply the numbers normally to get \(-18y\).
• With negative numbers, as you saw in the second multiplication, \(-6 \times -2x \) resulted in a positive \(12x\) as multiplying two negatives makes a positive.

It’s essential to follow the rules of multiplication consistently within algebra to solve problems correctly. This helps produce the desired and correct form of each term when tackled correctly.