Problem 29 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$3(2 x-4 y)-2(x+9 y)$$

Step-by-Step Solution

Answer

The simplified expression is \(4x - 30y\).

Step 1: Distribute the Multiplication

First, apply the distributive property to remove the parentheses. Multiply each term inside the parentheses by the outside multiplier. Start with the first expression: \[3(2x - 4y) = 3 \cdot 2x - 3 \cdot 4y = 6x - 12y.\] Now for the second expression: \[-2(x + 9y) = -2 \cdot x - 2 \cdot 9y = -2x - 18y.\]

Step 2: Combine Like Terms

Now that the parentheses are removed, combine like terms from the expressions: \[6x - 12y - 2x - 18y.\] This can be simplified by combining the 'x' terms and the 'y' terms separately:The 'x' terms: \[6x - 2x = 4x.\] The 'y' terms: \[-12y - 18y = -30y.\]

Step 3: Write the Simplified Expression

Write down the final simplified algebraic expression by combining the results from previous calculations:\[4x - 30y.\]

Key Concepts

Distributive Property

The distributive property is a key tool in algebra that allows us to simplify expressions by breaking them down into smaller, more manageable parts when dealing with multiplication and addition or subtraction. When you see an expression like \(a(b + c)\), the distributive property tells us we can simplify it by multiplying each term inside the parentheses by the term outside:

• Multiply \(a\) by \(b\).
• Multiply \(a\) by \(c\).

So, \(a(b + c) = ab + ac\).

In the original exercise, you can see this property in use with the expression \(3(2x - 4y)\). Here, we distribute the 3 to both \(2x\) and \(-4y\). This gives us \(6x - 12y\). Similarly, for the expression \(-2(x + 9y)\), you distribute \(-2\), resulting in \(-2x - 18y\).

Understanding how to spread the multiplication across terms inside parentheses is crucial as it helps us remove the parentheses and make the expression easier to work with.

Combining Like Terms

Once we have used the distributive property to remove parentheses, the next step is to combine like terms. Like terms are terms in an expression that have the same variable raised to the same power. For instance, \(6x\) and \(-2x\) are like terms because they contain the same variable \(x\).

To combine them, simply add or subtract their coefficients (the numerical part of the terms) while keeping the variable part intact. In the expression \(6x - 12y - 2x - 18y\) from the original exercise, follow these steps:

• Combine the \(x\) terms: \(6x - 2x = 4x\).
• Combine the \(y\) terms: \(-12y - 18y = -30y\).

Combining like terms is like cleaning up the clutter of an expression, making it more straightforward and easier to evaluate.

Simplifying Expressions

After applying the distributive property and combining like terms, the expression is in its simplest form. Simplifying expressions means making them easier to understand and use, often by transforming them into a more concise form.

In our case, after combining the like terms, the expression \(6x - 12y - 2x - 18y\) becomes \(4x - 30y\). This reduction of terms is what simplification entails. It involves ensuring there are no parentheses, like terms are combined, and there are no unnecessary components left in the expression.

Simplifying helps in identifying patterns or making further calculations more manageable. It lays the foundation for solving algebraic equations, making later steps in algebra more efficient and less error-prone.