Problem 38 - Algebra 2

Question

Simplify each expression. $$ 3(15 x-9 y)+5(4 y-x) $$

Step-by-Step Solution

Answer

The simplified expression is \(40x - 7y\).

Step 1: Apply the Distributive Property

Begin by distributing the coefficients in front of each parentheses. The expression starts as: \[ 3(15x - 9y) + 5(4y - x) \]Apply the distributive property:\[ 3 \times 15x - 3 \times 9y + 5 \times 4y - 5 \times x \]This simplifies to:\[ 45x - 27y + 20y - 5x \]

Step 2: Combine Like Terms

Now, combine the like terms in the expression. The expression after distribution is:\[ 45x - 27y + 20y - 5x \]First, combine the \(x\) terms:\[ 45x - 5x = 40x \]Next, combine the \(y\) terms:\[ -27y + 20y = -7y \]The expression now simplifies to:\[ 40x - 7y \]

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra that helps to eliminate parentheses by distributing the multiplication over addition or subtraction within an expression. This is especially useful when simplifying expressions with multiple terms. For example, when faced with an expression like \( 3(15x - 9y) + 5(4y - x) \), the distributive property tells us to multiply each term inside the parentheses by the coefficient outside.

Here's how this works step by step:

• For the expression \( 3(15x - 9y) \), distribute the 3 by multiplying it with both \( 15x \) and \( -9y \) to get \( 45x - 27y \).
• For \( 5(4y - x) \), distribute the 5 by multiplying it with both \( 4y \) and \( -x \) to obtain \( 20y - 5x \).

After applying the distributive property to every term, the expression transforms into \( 45x - 27y + 20y - 5x \). This step removes the parentheses, making the expression easier to work with in later stages.

Simplifying Expressions

Simplifying expressions is all about making them easier to understand and use. In algebra, this often means condensing an expression to a simpler form by performing operations such as addition, subtraction, and applying rules like the distributive property.

Once you've used the distributive property to get rid of parentheses, the next goal is to simplify the expression you've got. For example, after applying the distributive property to \( 3(15x - 9y) + 5(4y - x) \), you end up with:
\( 45x - 27y + 20y - 5x \).
This expression is not as simple as it could be because some terms can still be combined. This is where the next concept, combining like terms, comes into play. By simplifying the expression, you not only make calculations easier but also help to understand the relationships the expression describes more clearly.

Combining Like Terms

Combining like terms is a crucial step in simplifying expressions. This process reduces expressions down to fewer terms by adding or subtracting coefficients of identical variable parts. The idea is to group all the similar terms together and perform the necessary arithmetic operations.

In our working example, after applying the distributive property, we have the expression:
\( 45x - 27y + 20y - 5x \).
Here, you can see that there are multiple terms that can be combined because they have the same variable:

• Combine the \(x\) terms: \( 45x \) and \( -5x \). Adding these gives \( 40x \).
• Combine the \(y\) terms: \( -27y \) and \( 20y \). Adding these results in \( -7y \).

After combining like terms, the expression simplifies beautifully to \( 40x - 7y \).
This finished expression is much simpler and allows you to see relationships or solve equations more efficiently when dealing with similar expressions in algebraic contexts.