Problem 67 - Prealgebra

Question

Evaluate \(\frac{-4 y}{x}\) if \(x=2\) and \(y=-3\).

Step-by-Step Solution

Answer

The value of the expression is 6.

Step 1: Understand the Given Expression

We are given the expression \( \frac{-4y}{x} \), where we need to substitute the values of \( x \) and \( y \) provided in the problem to evaluate it.

Step 2: Substitute Known Values

Replace \( x \) with 2 and \( y \) with -3 in the expression \( \frac{-4y}{x} \). This gives us: \( \frac{-4(-3)}{2} \).

Step 3: Simplify the Numerator

Calculate \(-4 \times -3\). Multiplying these gives us \( 12 \), so now the expression is \( \frac{12}{2} \).

Step 4: Simplify the Expression

Divide 12 by 2, which results in 6.

Key Concepts

Substitution in Algebra

In algebra, substitution is a crucial technique used to simplify expressions by replacing variables with specific values. This method is especially useful in evaluating expressions when certain values are given for the variables involved. Let's break down how substitution works, using our exercise as an example.Imagine you have the algebraic expression \( \frac{-4y}{x} \). You're also given that \( x = 2 \) and \( y = -3 \). The goal of substitution is to plug these values into the expression wherever the variables appear. This process can be thought of in a few simple steps:

• Identify the variables in the expression (here, \( x \) and \( y \)).
• Replace each variable with the given value. So, substitute \( x \) with 2 and \( y \) with -3. This changes the expression to \( \frac{-4(-3)}{2} \).

This method effectively eliminates the variables, allowing you to work directly with numbers and move on to simplifying the expression.

Simplifying Expressions

After substituting the given values into the expression, the next step involves simplifying the expression to make it easier to understand and solve. In mathematics, simplifying means combining like terms and performing operations to reach a more straightforward form.For our task, after substitution, the expression becomes \( \frac{-4(-3)}{2} \). The first step in simplifying is focusing on the operation in the numerator, \(-4 \times -3\). Multiplying two negative numbers together gives a positive result. This is because, in mathematics, a negative times a negative equals a positive. The product of \(-4\) and \(-3\) is \(12\).Now the expression looks like \( \frac{12}{2} \). This is much simpler and ready for the final arithmetic operation, which is division.

Division in Algebra

Division in algebra is a fundamental operation that allows us to distribute a numerical value evenly across parts. It is the process of determining how many times a number fits into another number. In our example, once the expression is simplified to \( \frac{12}{2} \), we are left with the division step.To perform the division correctly, divide the numerator (12) by the denominator (2). In this case, 12 divided by 2 equals 6. Each unit of 2 fits into 12 exactly 6 times.Thus, after substituting the values and simplifying the expression properly, we determine that the evaluated expression equals 6. This approach ensures each operation is understood and carried out correctly, leading to accurate results.