Problem 104 - 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 \(\frac{x}{-y}\) for x=-42 and y=-7 is 6.

Step 1: Substituting Values

The first step in solving this problem is to substitute the given values of the variables into the expression. Thus the expression \(\frac{x}{-y}\) becomes \(\frac{-42}{-7}\).

Step 2: Simplification

In this step, divide -42 by -7. Remember that a negative divided by a negative yields a positive result. Therefore, \(\frac{-42}{-7} = 6\).

Key Concepts

Substituting Values

The process of substituting values in algebra involves replacing variables with their respective numerical values. It's a critical first step in evaluating expressions and provides a more concrete form for further operations.

Let's examine how to effectively substitute values using the original exercise we have, where the expression \(\frac{x}{-y}\) is provided, and we need to evaluate it for \(x=-42\) and \(y=-7\). To substitute correctly, simply replace each variable with the given number, ensuring to maintain the original signs as they are critical for the correct evaluation. Therefore, \(x\) becomes \(-42\), and \(-y\) becomes \(-(-7)\), which simplifies to \(+7\) since a double negative equals a positive.

Simplification in Algebra

Once the variables are substituted with their numerical values, the next step is simplification. Simplifying an algebraic expression means performing all possible arithmetic operations to reduce the expression to its simplest form.

In the context of our exercise, after substituting we get \(\frac{-42}{+7}\). To simplify this expression, divide \(-42\) by \(+7\). Simplification helps in distilling the expression to a single value that is easier to comprehend and further use in mathematical reasoning or applications. The goal is to always simplify as much as possible without changing the original value of the expression.

Negative Numbers Division

Dividing negative numbers can be a point of confusion, but it follows a simple rule: a negative number divided by another negative number yields a positive result. This rule is crucial and applies universally in mathematics, irrespective of the complexity of the expressions.

In our exercise, the division involved is \(-42\) divided by \(-7\). According to the rule, two negatives cancel each other out, leading to a positive result. Therefore, the expression simplifies to \(+6\), substantiating the fact that the quotient of two negative numbers is positive. This concept is not only fundamental in algebra but also forms the basis for understanding more advanced mathematical ideas.