Problem 61 - Algebra 1: Concepts and Skills

Question

Evaluate the expression when x 3 and y 6. $$ \frac{-8 x}{-4 y} $$

Step-by-Step Solution

Answer

The evaluated expression for the given \(x = 3\) and \(y = 6\) in \(-\frac{8x}{4y}\) is -1.

Step 1: Substitution

First, substitute \(x = 3\) and \(y = 6\) into the equation. So, the expression becomes \(-\frac{8*3}{4*6}\).

Step 2: Calculating the Numerator and Denominator

Calculate the numerator \(8*3\) to obtain 24 and the denominator \(4*6\) to obtain 24. So, the expression now reads as \(-\frac{24}{24}\).

Step 3: Simplifying the Expression

Finally, simplify the expression \(-\frac{24}{24}\) to find the result which equals -1.

Key Concepts

Substitution Method

The substitution method is an essential tool in algebra, especially when you're tasked with evaluating expressions. It involves replacing variables with their given numerical values. For instance, if an algebraic expression includes variables like 'x' and 'y,' and you're given that x is 3 and y is 6, you substitute these numbers into the original expression in place of the variables.

When applying this method correctly, as in our example where the original expression is \(\frac{-8x}{-4y}\), it is crucial to maintain the integrity of the expression's structure. After substituting x and y with 3 and 6, respectively, we get \(\frac{-8 \times 3}{-4 \times 6}\). This step requires precision and attention to ensure that each variable is replaced by its corresponding value without altering other parts of the expression.

Simplifying Expressions

Simplifying expressions is all about reducing an algebraic expression to its simplest form. The process makes it easier to understand and work with. After the substitution has been completed, we proceed to simplifying by performing the necessary arithmetic operations.

This can involve multiplying out the bracketed terms, canceling like terms, combining like terms, or other arithmetic operations. For the expression \(\frac{-8 \times 3}{-4 \times 6}\), we multiply the numbers in the numerator (top part of the fraction) and the denominator (bottom part of the fraction) separately to get \(\frac{-24}{-24}\). Since the negative signs cancel each other out, we're left with \(\frac{24}{24}\), which simplifies further to 1. The negative sign at the start applies to the whole fraction, making our result -1. This final simplification step is vital, as it gives us the most reduced form of the expression.

Numerical Expression

A numerical expression is a mathematical phrase that can consist of numbers, operations, and sometimes variables. In the context of our problem, after carrying out the substitution method and starting the simplification process, we arrive at a stage where only numbers and operations remain, with no variables in sight.

The numerical expression \(\frac{-24}{-24}\) doesn't include any variables and is ready for evaluation, which is the final step in solving these types of algebra problems. Evaluating the numerical expression leads us to a straightforward number, in this case, -1, which is the answer we're seeking. Understanding how to work with numerical expressions is key to solving algebraic equations because it allows us to simplify complex problems into a form where we can easily find a solution.