Problem 102 - Prealgebra and Introductory Algebra

Question

Evaluate the expression for the given values of the variables. $$\frac{x}{y}, \text { for } x=-42 \text { and } y=-7$$

Step-by-Step Solution

Answer

The value of the expression for the provided values is 6.

Step 1: Understand the Problem

The given expression is \( \frac{x}{y} \). The problem is asking you to find the value of this expression when \( x=-42 \) and \( y=-7 \).

Step 2: Substitute Values

Substitute the given values into the expression. Replace every instance of \( x \) with -42 and \( y \) with -7, so the expression will become \( \frac{-42}{-7} \).

Step 3: Solve The Expression

Perform the division operation in the expression \( \frac{-42}{-7} \). It gives 6 as the result, because a negative number divided by another negative number results in a positive number.

Key Concepts

Substitution

Substitution in algebra involves replacing variables with specific values. In the exercise, the expression is \( \frac{x}{y} \). You replace \( x \) with -42 and \( y \) with -7. After substitution, the equation becomes \( \frac{-42}{-7} \).

Here's why substitution is key:

• It transforms abstract expressions into solvable arithmetic problems.
• Helps us fully utilize given data to find solutions.
• Makes complex algebraic expressions more tangible.

When substituting, be careful to maintain the original structure of the expression, substituting values only where indicated. This accuracy ensures correct computation of the final result.

Division in Algebra

Division is a fundamental operation in algebra, simplifying expressions where one quantity is divided by another. In this particular exercise, we evaluate \( \frac{-42}{-7} \).

Important aspects of division in algebra:

• Quotient Interpretation: A division like \( \frac{a}{b} \) identifies how many times \( b \) fits into \( a \).
• Maintaining integer relations: When possible, division of integers should result in integers.
• Simplification: It reduces the complexity of expressions, often transforming them into more manageable forms.

In the example, dividing -42 by -7 yields 6, highlighting how division reduces negative numbers to positive outcomes when both dividend and divisor are negative.

Negative Numbers

Negative numbers can seem tricky but are crucial in algebra. They represent values less than zero, and dealing with them involves specific rules.

Consider these key points:

• Multiplication and Division Rules: The product or quotient of two negative numbers is positive.
• Direction and Position: Negative numbers are located to the left of zero on a number line.
• Real-world applications: Useful for denoting debts, decreases, or values below a certain threshold.

For instance, dividing \( -42 \) by \( -7 \) results in 6, illustrating how the division of two alike signs transforms the negative signs into a positive quotient. Remembering this rule simplifies working with negative numbers across all mathematical operations.