Problem 27 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The expression \(5(y-2)\) without parentheses is \(5y - 10\).

Step 1: Identify the Variables

In the expression \(5(y-2)\), 5 is the number that will be distributed to every term in the parentheses, y and -2. The individual elements can be identified as follows: a = 5, b = y, c = -2.

Step 2: Apply the Distributive Property

According to the Distributive Property, you will multiply 'a' by 'b' and 'a' by 'c'. Do the computation: 5 * y = 5y and 5 * -2 = -10.

Step 3: Put Together the Results

After the multiplication results obtained from the previous step, you place them back into the equation like so: \(5y - 10\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a mathematical way to represent real-world situations using letters, numbers, and arithmetic operations. They consist of variables, constants, and combinations of these through operations like addition, subtraction, multiplication, and division. In the expression \(5(y-2)\), \(y\) is the variable representing an unknown quantity, and \(-2\) is the constant.
Key Components of Algebraic Expressions:

• Variables: Symbols that stand in for unknown values. In our expression, \(y\) is the variable.
• Constants: Numbers on their own. These do not change. Here, \(-2\) is the constant.
• Coefficients: Numbers multiplied by variables. In \(5y\), 5 is the coefficient of \(y\).
• Terms: Parts of an expression separated by plus or minus signs. \(5y\) and \(-10\) are terms in the final expression.

Multiplication

Multiplication in algebra is the process of scaling one number by another. It's a fundamental arithmetic operation used to increase the value of a number proportionally. When employing multiplication with variables, it’s important to consider both the numerical and variable components.
Performing Multiplication in Algebra:

• Scalar Multiplication: A constant multiplies a variable and changes the value of the variable proportionally. For instance, 5 multiplied by \(y\) results in \(5y\).
• Sign Consideration: Pay attention to positive and negative signs. Multiply 5 by \(-2\) to get \(-10\).
• Associative Property: Enables re-grouping and does not affect the product. That is, \((5 \times y) \times -2 = 5 \times (y \times -2)\).

These techniques allow multiplication to be seamlessly integrated within algebraic expressions, enhancing our problem-solving capabilities.

Mathematical Properties

Mathematical properties make algebra easier and more systematic. One such property widely used in algebra is the Distributive Property which is essentially linking multiplication and addition/subtraction.
The Distributive Property Explained:

• The property states that for any numbers or variables \(a\), \(b\), and \(c\), the equation holds: \(a(b + c) = ab + ac\).
• It allows you to "distribute" a number outside the parenthesis to each term inside. For instance, in \(5(y - 2)\), 5 is multiplied by both \(y\) and \(-2\).
• This property simplifies expressions and is an essential tool in algebra and can help solve complex equations more easily.

Understanding and using the Distributive Property assists in the simplification of expressions and ensures accuracy in computations.