Problem 111 - Beginning and Intermediate Algebra

Question

If \(x=-5\) and \(y=-3,\) evaluate each expression. $$ \frac{-3-y}{x-4} $$

Step-by-Step Solution

Answer

The evaluated expression is 0.

Step 1: Substitute the values for x and y

Replace x with -5 and y with -3 in the expression. The expression becomes: \(-3 - (-3)\) in the numerator and \(-5 - 4\) in the denominator. This results in: \(\frac{-3 - (-3)}{-5 - 4}\).

Step 2: Simplify the numerator

Simplify \(-3 - (-3)\). Since subtracting a negative is equivalent to adding the positive, it becomes: \(-3 + 3 = 0\). The numerator is now 0.

Step 3: Simplify the denominator

Simplify \(-5 - 4\). Combine the terms to get: \(-5 - 4 = -9\). The denominator is now -9.

Step 4: Evaluate the expression

The expression is now \(\frac{0}{-9}\). Any number divided by another number, except zero, is zero. Therefore, the result is 0.

Key Concepts

Numerator and Denominator

In any fraction, the numerator and denominator are two crucial components. The numerator is the top part of the fraction, and it indicates how many parts we have. The denominator, on the other hand, is the bottom part of the fraction. It shows the number of equal parts the whole is divided into.

Using the original expression \[\frac{-3-y}{x-4}, \]we identify \(-3-y\) as the numerator and \(x-4\) as the denominator.

• The numerator tells us the amount or value of interest in this fraction.
• The denominator helps determine the scale or division by which the numerator is measured.

The relationship between these two parts is essential in evaluating and simplifying expressions, allowing us to understand the division and resulting values.

Substitution in Expressions

Substitution involves replacing variables with given values or numbers to reduce expressions. When tackling algebraic expressions, it helps simplify and evaluate them effectively.

Let's explore this with the expression \(\frac{-3-y}{x-4} \) when we know \(x=-5\) and \(y=-3\). By substituting these values, we can transform the expression as follows:

• In the numerator: replace \(y\) with \(-3\), to get \(-3 - (-3)\).
• In the denominator: replace \(x\) with \(-5\), to become \(-5 - 4\).

Substitution turns abstract expressions into numbers, ready to be worked with, leading us closer to the solution.

Simplifying Algebraic Expressions

Simplifying an algebraic expression means reducing it to its most basic form. This often involves combining like terms and carrying out basic arithmetic operations.

In our current context, the expression went from \(\frac{-3 - (-3)}{-5 - 4} \) to \(\frac{0}{-9}\) through simplification:

• In the numerator: the expression \(-3 - (-3)\) simplifies to \(0\). Here, subtracting a negative is effectively adding the positive.
• In the denominator: \(-5 - 4\) simplifies to \(-9\), by adding two negative numbers, resulting in a deeper negative value.

When you simplify a fraction this way, if the numerator becomes \(0\), the whole fraction represents \(0\). Simplification is about eliminating excess and making calculations straightforward, focusing on the clearest expression of a value possible.