Problem 31 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$9-4(9+y)$$

Step-by-Step Solution

Answer

The simplified expression is \(-27 - 4y\)

Step 1: Simplify Inside Brackets

The first step in simplifying the given operation is to perform the multiplication inside the parenthesis. So, the expression becomes \(9 - 4 \times 9 - 4 \times y\).

Step 2: Perform the multiplication

Afterwards, perform the multiplication: \(9 - 36 - 4y\)

Step 3: Simplify the expression

Finally, simplify the expression by performing the subtraction operation: \(-27 - 4y\)

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions involves breaking them down into the simplest form, making them easier to work with. The key is to consolidate like terms and clear out unnecessary groupings, like parentheses, without altering the value of the expression. In our example, start by looking inside the parentheses or brackets. This is often where simplification begins, especially if there are operations to perform.

• Start by handling operations inside the parentheses.
• Apply any necessary mathematical operations like multiplication or division.
• Combine like terms, if possible, to reduce redundancy in the terms.

This means, for the given expression \(9 - 4(9+y)\), you'll want to manage the terms inside to enable easier combinations or cancellations later.

Distributive Property

The distributive property is a fundamental principle in algebra that allows you to simplify expressions in which multiplication is distributed across terms inside parentheses. It is particularly useful when dealing with expressions involving both addition (or subtraction) and multiplication.
The property is mathematically expressed as \(a(b + c) = ab + ac\), meaning that a factor outside the parentheses is applied to each term inside.
In our example, the distributive property tells us to take \(-4\) and multiply it by both \(9\) and \(y\). Here, you proceed by calculating \(-4 \times 9\) to get \(-36\), and \(-4 \times y\) to get \(-4y\). This results in the new expression \(9 - 36 - 4y\).

• Helps to eliminate parentheses by distributing a multiplier through grouped terms.
• Keeps related operations organized across multiple terms.
• Enables straightforward simplification of complex expressions.

Order of Operations

The order of operations is critical to ensuring accurate calculations in algebraic expressions. It's like following a set of instructions, ensuring that every step is done in the correct sequence. The standard rule is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
For the expression \(9 - 4(9+y)\) the order of operations clarifies that:

• The first step is to simplify inside the parentheses, as it contains terms added together.
• Next is the multiplication with \(-4\), applying the distributive property as discussed.
• Finally, you'll complete any remaining addition or subtraction operations in the expression, from left to right.

In our example, after using the distributive property, the subtraction is performed linearly from left to right, reducing \(9 - 36 - 4y\) to \(-27 - 4y\). Each step reinforces the correctness and simplicity of your solution.