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Chapter 6: Problem 14

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

Short Answer

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

Step by step solution

01

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 \( y - 4 \).
02

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 \( y - 4 \).
03

Find the Necessary Multiplier

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Solve equation. If a solution is extraneous, so indicate. \(\frac{z+2}{z+8}=\frac{z-3}{z-2}\)

For each expression in part (a), perform the indicated operations and then simplify, if possible. Solve equation in part (b) and check the result. a. \(\frac{m}{m-2}-\frac{1}{m-3}\) b. \(\frac{m}{m-2}-\frac{1}{m-3}=1\)

Solve equation. If a solution is extraneous, so indicate. \(\frac{5}{x+4}+\frac{1}{x+4}=x-1\)

Let \(Q(x)=x^{4}-3 x^{3}+2 x^{2}+x-3 .\) Evaluate \(Q(x)\) by substituting the given value of \(x\) into the polynomial and simplifying. Then evaluate the polynomial by using the remainder theorem and synthetic division. See Example 4. $$ Q(1) $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{-10}{t+3}=1-\frac{11}{t-3}\)
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