Problem 53 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ 10-3(2 x+3 y) $$

Step-by-Step Solution

Answer

The simplified expression is \(10 - 6x - 9y\).

Step 1: Identify the Expression Inside the Parentheses

First, look at the expression inside the parentheses: \(2x + 3y\). We need to distribute -3 across this expression.

Step 2: Distribute the Value Outside the Parentheses

Multiply each term inside the parentheses by -3. This gives us: \(-3 \times 2x = -6x\) and \(-3 \times 3y = -9y\).

Step 3: Rewrite the Expression Without Parentheses

After distributing, rewrite the expression: \(10 - 6x - 9y\).

Step 4: Combine Like Terms

Examine the expression. There are no like terms to combine, so the simplified expression is \(10 - 6x - 9y\).

Key Concepts

Understanding the Distributive Property

The distributive property is a vital concept in algebra that allows us to simplify expressions by removing parentheses. It indicates that multiplying a number by a group of terms inside the parentheses is the same as doing each multiplication individually. This property can be expressed as \( a(b + c) = ab + ac \). Essentially, you distribute the number outside the parentheses to each term within them.

In our example, the expression is \( 10 - 3(2x + 3y) \). By applying the distributive property, we will multiply \(-3\) with \(2x\) and \(3y\), which means:

• \(-3 \times 2x = -6x\)
• \(-3 \times 3y = -9y\)

This step is crucial as it helps to eliminate the parentheses and set up the expression for further simplification.

Exploring Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They form the building blocks of algebra that you will frequently encounter and manipulate. In our exercise, the expression \(2x + 3y\) is algebraic.

• "2x" signifies that the variable \(x\) is multiplied by 2.
• "3y" indicates that the variable \(y\) is multiplied by 3.

Understanding these fundamental components is essential for operations like distribution. Recognizing each term within an algebraic expression lets you apply mathematical properties correctly and efficiently. Always pay attention to the signs, coefficients, and variables as you work through problems.

Mastering Parentheses Removal

Removing parentheses is a key algebraic skill used to simplify expressions. It involves re-writing expressions by eliminating the parentheses while maintaining equivalence. This is often done using the distributive property.

In the original problem, we started with \(10 - 3(2x + 3y)\). By applying the distributive property and multiplying \(-3\) by each term within the parentheses, the parentheses are effectively eliminated:

• The term \(-3 \times 2x = -6x\)
• The term \(-3 \times 3y = -9y\)

Thus, the expression becomes \(10 - 6x - 9y\), with no more parentheses to manage. Parentheses removal simplifies the expression and often reveals like terms that can be combined, although in this case, there are none. Always ensure the final expression is as simplified as possible.