Problem 36 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ 5(7+8 y) $$

Step-by-Step Solution

Answer

The expression simplifies to \(35 + 40y\).

Step 1: Apply the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), the expression \(a(b + c)\) can be expanded to \(ab + ac\). Here, we have \(5(7 + 8y)\). Use the distributive property to distribute \(5\) to both terms inside the parentheses: \[ 5 \cdot 7 + 5 \cdot 8y. \]

Step 2: Perform the Multiplications

Now, perform the multiplications from Step 1. Calculate \(5 \times 7\) which equals \(35\), and \(5 \times 8y\) which equals \(40y\). So the expression becomes: \[ 35 + 40y. \]

Step 3: Simplify the Expression

The expression \(35 + 40y\) is already simplified because there are no like terms to combine. Therefore, the simplification process ends here.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. They are like mathematical phrases that help us describe relationships between different values. In our example, the expression is given as \(5(7+8y)\). Here, 7 and 8 are numbers, while \(y\) is a variable which represents an unknown value. Variables are used in algebra to generalize mathematical problems and create formulas. The multiplication of 5 with the entire expression inside the parentheses involves the use of the distributive property. Understanding how to handle algebraic expressions is crucial to solving equations and understanding more complex math topics.

Simplifying Expressions

Simplifying expressions is all about making an expression easier to understand and work with. It often involves combining like terms, which are terms that have the same variable raised to the same power, thereby reducing the complexity. In the expression \(35 + 40y\), \(35\) is a constant term, and \(40y\) is a term with a variable. Typically, simplifying expressions means performing operations where possible, such as addition or subtraction.

• Check for like terms and combine them.
• Carry out any possible arithmetic operations.

However, in our example, there are no like terms to combine as \(35\) and \(40y\) are different. Hence, the expression remains the same after simplification. Knowing when an expression is fully simplified is an important skill when working with algebra.

Mathematical Operations and the Distributive Property

Mathematical operations include addition, subtraction, multiplication, and division. The distributive property is an essential algebraic property that is used to multiply a single term across two or more terms within parentheses. In simpler terms, when you see an expression like \(a(b + c)\), you can distribute \(a\) by multiplying it with each term inside the parentheses separately. This makes it easier to solve expressions by breaking them down.

• Multiply the numbers outside by each term inside the parentheses.
• Combine the results as a simpler expression.

For example, with \(5(7 + 8y)\), we distribute 5 to both 7 and \(8y\), resulting in \(35 + 40y\). This simplifies solving algebraic problems and is frequently used in various areas of mathematics. Understanding this property helps in both grasping algebraic concepts and solving equations effectively.