Problem 105 - 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 solution to the given expression is 6.

Step 1: Replace the variables with the given values

First, replace 'x' with -42 and 'y' with -7, in the equation \(-x / -y\), yielding \(-( -42) / ( -7)\). The minus sign before each variable is part of the original expression. It's important not to forget them during the substitution.

Step 2: Simplify the expression

After the substitution, we have positive 42 divided by positive 7, because two negatives make a positive. So, the equation simplifies to \(42 / 7\).

Step 3: Perform the division

Finally, divide 42 by 7, which results in 6.

Key Concepts

Variable Substitution

Variable substitution involves replacing variables in an expression with numerical values provided. This process allows us to evaluate expressions. For the expression \(\frac{-x}{-y}\), with \(x = -42\) and \(y = -7\), substitution is as follows:

• Replace \(x\) with \(-42\), making the term \(-(-42)\).
• Replace \(y\) with \(-7\), resulting in the term \(-(-7)\).

This step is crucial because it transforms the variable-based expression into a numerical one. This makes the expression easier to evaluate and simplifies further calculations. Be careful to include all negative signs from the original expression to ensure accuracy in solving the problem.

Simplification Steps

After substituting the variables, it is often necessary to simplify the expression. Simplification is the process of making an equation easier to work with by combining like terms or reducing expressions. An important part here is understanding how negative signs interact:

• When substituting \(-(-42)\), the two negatives cancel each other out, turning it into \(42\).
• Similarly, \(-(-7)\) becomes \(7\).

The equation \(\frac{-(-42)}{-(-7)}\) simplifies to \(\frac{42}{7}\). Understanding that two negative signs result in a positive number is key. This simplification sets up the expression for easy division in the next step.

Division

Once the expression is simplified into a form like \(\frac{42}{7}\), you can proceed with division. Division is the process of finding how many times one number is contained within another.

• Here, \(42\) is divided by \(7\).
• This calculation is straightforward: \(42 \div 7 = 6\).

Performing division between two numbers can be done by counting how many times the divisor (7) can fit into the dividend (42). This results in the final answer. Make sure the division is performed correctly to avoid errors.

Negative Numbers

Negative numbers can change the sign of an expression when they appear twice. Understanding how they work is crucial in evaluating expressions like \(\frac{-x}{-y}\).

• Two negative numbers multiplied or divided together result in a positive number.
• In the expression \(-(-42)\) and \(-(-7)\), two negatives cancel each other out, producing positive numbers.

This rule is fundamental when simplifying expressions involving negative numbers. It is a common point of confusion, so practicing with different expressions can help reinforce this concept. Recognizing these relationships makes solving related math problems easier.