Problem 110 - Beginning and Intermediate Algebra

Question

If \(x=-5\) and \(y=-3,\) evaluate each expression. $$ \frac{2 y-12}{x-4} $$

Step-by-Step Solution

Answer

The value of the expression is 2.

Step 1: Substitute the Variables

Given that \(x = -5\) and \(y = -3\), substitute these values into the expression. We start with the original expression: \(\frac{2y - 12}{x - 4}\). Substitute to get \(\frac{2(-3) - 12}{-5 - 4}\).

Step 2: Simplify the Numerator

Simplify the expression in the numerator: \(2(-3) = -6\). Therefore, the numerator becomes \(-6 - 12\). Simplify this further to get \(-18\).

Step 3: Simplify the Denominator

Simplify the expression in the denominator: \(-5 - 4\) which gives \(-9\).

Step 4: Evaluate the Fraction

Combine the simplified numerator and denominator to evaluate the expression: \(\frac{-18}{-9}\). Simplify the fraction to get \(2\).

Key Concepts

Substitution Method

The substitution method helps simplify mathematical expressions by replacing variables with known values. This technique is widely used not only in algebra but also in calculus and other areas of mathematics.
Here's how it works:

• Identify the variables in the expression that need to be substituted.
• Replace each variable with its given value.

For example, in the given expression \(\frac{2y - 12}{x - 4}\), we substitute \(x=-5\) and \(y=-3\) into the expression. This results in \(\frac{2(-3) - 12}{-5 - 4}\). Substitution transforms complicated expressions into simpler numerical ones that are easier to evaluate.

Numerator and Denominator

When working with fractions, it is crucial to understand the parts of the fraction: the numerator and the denominator. These components play a fundamental role in fraction operations like addition, subtraction, multiplication, and division.

• The numerator is the top part of a fraction and indicates how many parts we have.
• The denominator is the bottom part of a fraction and shows the total number of equal parts.

The expression \(\frac{2y - 12}{x - 4}\) has \(2y - 12\) as the numerator and \(x - 4\) as the denominator. After performing the substitutions, the simplified forms become \(-18\) and \(-9\) respectively.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, which means the numerator and the denominator have no common factors other than 1. This process makes the fraction more interpretable and easier to work with.
To simplify a fraction:

• Divide both the numerator and the denominator by their greatest common divisor (GCD).

In our example, after substitution and simplification, we get \(\frac{-18}{-9}\).
Both \(-18\) and \(-9\) are divisible by \(-9\). Therefore, the fraction simplifies to \(\frac{2}{1}\), or simply \(2\). This means the original complex-looking expression evaluates to a straightforward number.