Problem 45 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ (y+9)(-1) $$

Step-by-Step Solution

Answer

The expression \( (y+9)(-1) \) can be rewritten as \( -y - 9 \) using the distributive property.

Step 1: Apply the Distributive Property

The distributive property states that for any numbers a, b, and c, \( a*(b+c) = a*b + a*c \). Here, all we need to do is apply this rule. Hence, our expression \( (y+9)*(-1) \) can be rewritten as \( -1*y + (-1)*9 \).

Step 2: Simplify the multiplication

Now we simplify each of the terms individually. So, \( -1*y \) simplifies to just \( -y \) and \( (-1)*9 \) simplifies to \( -9 \).

Key Concepts

Algebraic Expressions

Algebraic expressions are the cornerstones of algebra, composed of numbers, variables, and arithmetic operations. An algebraic expression can range from a simple variable like \( y \) to more complex combinations such as \( y + 9 \). Here, the expression \( y + 9 \) includes a variable \( y \) and a constant \( 9 \), linked by addition.

When dealing with algebraic expressions, it is crucial to understand how to manipulate and rearrange them using various algebraic properties, such as the distributive property. Mastery of these concepts will allow you to solve equations and simplify expressions efficiently. Simplification often involves removing parentheses, combining like terms, and reducing expressions to make them easier to work with. This particular exercise highlights the use of the distributive property, a fundamental tool in algebra to help reorganize expressions.

Simplification

Simplification in mathematics refers to the process of breaking down expressions into their most straightforward form. In the exercise, simplifying the expression involves two main steps: applying the distributive property and then performing basic arithmetic operations.

With the expression \((y+9)(-1)\), simplification begins with distributing \(-1\) to both terms inside the parentheses, following the rule:

• Multiply \(-1\) by \(y\)
• Multiply \(-1\) by \(9\)

This yields \(-1 \, \cdot \, y + (-1) \, \cdot \, 9 = -y - 9\). Each term in the resulting expression is the product of multiplication, simplified from the original, more complex expression.

Understanding how to simplify expressions allows students to more easily solve algebraic equations. This exercise reinforces the fundamental technique of applying the distributive property to challenges involving parentheses.

Negative Numbers

Negative numbers represent values less than zero and have unique properties in arithmetic operations. When combined with positive numbers or other negative numbers, they require careful handling to avoid errors.

For instance, in this exercise, the number \(-1\) is worked with under the distributive property. Consider the properties and rules of negative numbers during multiplication:

• Multiplying a positive number by a negative number results in a negative product: \(-1 \, \cdot \, y = -y\).
• Similarly, multiplying a positive number by a negative number yields a negative result: \((-1) \, \cdot \, 9 = -9\).

These rules ensure that the expression is correctly simplified.

Handling negative numbers correctly is essential not only in basic arithmetic but also in larger algebraic contexts, as seen in simplification and solving equations.