Problem 63 - Prealgebra

Question

Simplify each expression. $$-9(y-4)$$

Step-by-Step Solution

Answer

The simplified expression is \(-9y + 36\).

Step 1: Distribute the Negative Sign

We begin by distributing the negative sign, along with the factor -9, to each term inside the parentheses. This means multiplying each term in the parentheses by -9. So, we have: \[-9(y - 4) = -9 \cdot y - 9 \cdot (-4)\].

Step 2: Perform the Multiplications

Next, perform the multiplication operations: - \(-9 \times y = -9y\), - \(-9 \times -4 = 36\).Therefore, substituting these values back, we have \(-9y + 36\).

Step 3: Final Simplified Expression

Combine the results from the multiplication to obtain the final simplified expression, which is \(-9y + 36\).

Key Concepts

Simplification

Simplification in algebra is like tidying up a messy room. We aim to make the algebraic expressions as straightforward as possible. In the exercise, we started with the expression \(-9(y-4)\). Here, simplification involves using specific algebraic rules to reduce the expression into a simpler form.

The goal of simplification is to make it easier to understand and solve equations or compare expressions. In our case, through simplification, we turned the given expression into \(-9y + 36\). To achieve this, we used algebraic techniques like distributing numbers across terms (discussed in the next section).

Remember, when simplifying expressions:

• Combine like terms, which means adding or subtracting similar variables, such as \(2x + 3x\).
• Use operations like addition, subtraction, multiplication, or division logically and carefully.
• Focus on following the order of operations (PEMDAS/BODMAS).

The effort in simplification lies in making complex expressions neat and useful for further analysis or computation.

Distributive Property

The distributive property is a crucial concept that helps us simplify expressions in algebra. It allows you to "distribute" a factor outside the parentheses to each term within the parentheses. In our exercise, we applied this property to simplify \(-9(y-4)\).

The rule here is: \(a(b + c) = ab + ac\).

Let's break our example down:

• We had \(-9(y - 4)\). Here, \(-9\) is multiplied by each term inside the parentheses.
• This gives us: \(-9 \cdot y\) and \(-9 \cdot (-4)\).
• Solving these multiplications, we got \(-9y\) and \(36\).

This fundamental tool in algebra allows for simplifying expressions, solving equations, and finding products of numbers or variables effectively, even with more complex expressions.

Negative Numbers

Understanding negative numbers can sometimes be tricky but is essential for algebraic operations. In algebra, negative numbers work just like positive numbers, but in the opposite direction along the number line.

When dealing with negative numbers, keep these points in mind:

• Adding a negative number is the same as subtracting a positive number (e.g., \(5 + (-3) = 2\)).
• Subtracting a negative number is akin to adding its positive counterpart (e.g., \(5 - (-3) = 8\)).
• Multiplying two negative numbers results in a positive product (e.g., \(-2 \times -3 = 6\)).
• Multiplying a positive and a negative number yields a negative product (e.g., \(2 \times -3 = -6\)).

Specifically in our exercise, by distributing \(-9\), we handled negative numbers, performing multiplications \(-9 \times y\) resulting in \(-9y\) and \(-9 \times -4\) giving a positive \(36\). Understanding these rules ensures accuracy in algebraic simplifications.