Problem 14 - Intermediate Algebra

Question

By what must \(y-4\) be multiplied to obtain \(4-y ?\)

Step-by-Step Solution

Answer

\( y - 4 \) must be multiplied by \(-1\) to obtain \(4 - y\).

Step 1: Recognize Expression Patterns

First, let's observe the expressions \( y - 4 \) and \( 4 - y \). Notice that they look very similar, but they have different signs for their terms. Specifically, \( 4 - y \) is the negative of \( y - 4 \).

Step 2: Determine the Sign Relationship

To transform \( y - 4 \) into \( 4 - y \), we can write: \( 4 - y = -(y - 4) \). This shows that \( 4 - y \) is simply the negative of \( y - 4 \).

Step 3: Find the Necessary Multiplier

Since \( 4 - y = -(y - 4) \), the expression \( y - 4 \) must be multiplied by \(-1\) to obtain \(4 - y\).

Key Concepts

Negative of expressions

In algebra, expressions like \( y - 4 \) and \( 4 - y \) may look similar. However, their structures and signs differ. A negative of an expression inverts the sign of all its terms. For instance, let's take \( y - 4 \). If we want to find its negative, we have to switch the sign of each term within it. Instead of subtracting 4, we add 4, and instead of subtracting \( y \), we add \( y \). This transformation turns \( y - 4 \) into \( 4 - y \).

Recognizing these patterns makes manipulating algebraic expressions easier. Expressing one as the negative of another helps identify what needs to change in the expression.

Sign relationships in algebra

Algebra involves understanding how signs interact within expressions. Signs determine how terms relate to each other. Let's explore the patterns in our original expressions, \( y - 4 \) and \( 4 - y \):

• \( y - 4 \) has a positive \( y \) and a negative 4.
• \( 4 - y \) has a positive 4 and a negative \( y \).

To change one into another, we need to reverse the signs of both terms. This change illustrates the concept of subtracting the whole expression.

If you're ever unsure about signs, remember: flipping signs means multiplying the entire expression by \(-1\). Keeping track of signs is crucial in algebra, especially when solving equations and simplifying expressions.

Multiplication with negative numbers

Multiplication with negative numbers changes the sign of the product. Here's how it works:

• Multiplying a positive number by a negative gives a negative result.
• Multiplying two negative numbers yields a positive result.

When asked to find out how to transform \( y - 4 \) into \( 4 - y \), we multiply by \-1\. This is because the expression \( 4-y \) is the negative of \( y-4 \). Multiplying an expression by \-1\ switches all the signs in the expression.

For example, if \( a \) is positive, \-a\ is negative, and vice versa. This sign flip is a fundamental part of algebra. Understanding how negative multiplications impact expressions helps in simplifying problems.