Problem 75 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the expressions. $$ (x+y)(4 a+3 b) $$

Step-by-Step Solution

Answer

Answer: The expanded form of the expression \((x+y)(4a+3b)\) is \(4ax + 3bx + 4ay + 3by\).

Step 1: Apply the distributive property to the first term

First, we will distribute \((x+y)\) to both terms inside the parentheses \((4a+3b)\). Using the distributive property, this results in: $$ (x+y)(4a+3b) = x(4a+3b) + y(4a+3b) $$

Step 2: Apply the distributive property again

Now, we'll apply the distributive property again on the two expressions inside the parentheses to get: $$ x(4a+3b) = 4ax + 3bx $$ and $$ y(4a+3b) = 4ay + 3by $$

Step 3: Combine the expressions

Finally, combine the expressions from Step 2 to get the expanded form of the given expression: $$ (x+y)(4a+3b) = (4ax + 3bx) + (4ay + 3by) $$ Our final answer is: $$ (x+y)(4a+3b) = 4ax + 3bx + 4ay + 3by $$

Key Concepts

Expanding Expressions

When you're faced with the task of expanding algebraic expressions, the key idea is to remove the parentheses and express the formula as a sum of simpler parts. This is where the Distributive Property comes into play. In the expression \((x+y)(4a+3b)\), expansion involves applying this property to each term.

To begin expanding, we distribute the terms inside the parentheses. Distribute \((x+y)\) across \((4a+3b)\) by multiplying each term separately:

• \(x\) is multiplied by both \(4a\) and \(3b\).
• \(y\) is also multiplied by both \(4a\) and \(3b\).

This results in a combination of simpler expressions, making it easier to understand and solve in future steps.

The process concludes with adding these expressions together, resulting in a fully expanded form. This approach not only simplifies expressions but also maintains their equivalency with the original expression.

Algebraic Expressions

Algebraic expressions can be thought of as a group of numbers, letters, and mathematical symbols assembled to represent a quantity or a relation. They often contain variables (such as \(x\) or \(y\)) that can take on different values to influence the expression's outcome. Let's consider our specific example of the expression \((x+y)(4a+3b)\).

In this algebraic expression:

• \(x\) and \(y\) are variables, representing unknown values that can change depending on different factors.
• \(4a\) and \(3b\) are terms that are themselves algebraic expressions, composed of constants \(4\) and \(3\), multiplied by variables \(a\) and \(b\), respectively.

Understanding each component of an algebraic expression helps in efficiently solving, simplifying, and expanding it. Recognizing the terms and how they relate through operations is crucial to mastering algebra.

Mathematical Operations

Mathematical operations are fundamental processes that we perform on numbers and algebraic expressions. In the case of expanding expressions, multiplication plays a central role through the Distributive Property. The operations we focus on involve:

• Multiplication: Combine expressions by distributing terms. This is seen when we multiply \((x+y)\) across \((4a+3b)\).
• Addition: After multiplication, the results are summed to create the expanded expression.

Mastering these operations allows for deeper comprehension and the ability to manipulate expressions with confidence. The Distributive Property itself is a specific application of multiplication and addition, showing how these fundamental operations work together to simplify and expand algebraic expressions effectively.