Problem 29 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$5(y-2)$$

Step-by-Step Solution

Answer

The simplification of the expression \(5(y - 2)\) using the distributive property results in \(5y - 10\).

Step 1: Define the Variables

From the given expression, \(5(y - 2)\), a = 5, b = y and c = 2.

Step 2: Apply Distributive Property

Now applying the distributive property \(a*(b - c) = ab - ac\), the expression becomes \(5*y - 5*2\)

Step 3: Simplify Expression

Simplify the expression to \(5y - 10\)

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. In the expression \(5(y-2)\), we see both numbers and a variable, \(y\). Variables in algebra stand for unknown values and, when combined with numbers and operations, form expressions that can represent many different situations.
For example, \(5(y-2)\) can model a real-world scenario, like calculating the total cost when buying \(y\) items, each \$5, deducting some fixed amount like a discount represented by 2. The expression captures relationships between quantities.
Understanding how algebraic expressions work is crucial because they are the foundation of algebra. They can be rearranged and solved to find the value of variables or to simplify complex problems.

Simplification

Simplifying an expression involves combining or reducing terms to make the expression easier to read or solve. This process makes expressions more manageable, especially when they will be part of more significant equations or problems.
In the exercise, after using the distributive property, the expression \(5(y-2)\) transformed into \(5y - 10\). Simplification here involves performing the multiplication and subtraction to remove the parentheses and present the expression in its simplest form.

• Multiply 5 by \(y\) resulting in \(5y\).
• Then, multiply 5 by 2, getting 10.
• Combine these results to form the simplified expression, \(5y - 10\).

Simplification does not change the original value but transforms how it is represented. It is an essential skill in algebra that allows one to work effectively with expressions and equations.

Properties of Operations

Properties of operations help in manipulating and simplifying expressions. One of the most vital properties is the distributive property, used in our exercise to structure the expression without parentheses.
The distributive property allows you to express \(a(b - c)\) as \(ab - ac\). This means that each term inside the parentheses is multiplied by the outside term, distributing multiplication over addition or subtraction.

• In \(5(y-2)\), the 5 multiplies both \(y\) and 2.
• This gives \(5y - 5 \times 2\).
• Thus, we arrive at \(5y - 10\).

These properties are tools in algebra for restructuring and simplifying expressions and equations. Understanding them allows for more flexible and efficient problem-solving, particularly as expressions or equations become more complex in higher-level math.