Question

Evaluate the expression. $$ \frac{x+y}{12} \text { when } x=-24 \text { and } y=6 $$

Short answer

The result of the expression when \(x=-24\) and \(y=6\) is -1.5

Step-by-step solution

Substitute variables

First, substitute the given values of x and y into the expression. So the expression becomes: \(\frac{(-24)+6}{12}\).

Simplify Numerator

Next, the numerator of the expression needs to be simplified: \(\frac{(-24 + 6)}{12} \) resolves to \(\frac{-18}{12}\).

Simplify Fraction

The fraction now needs to be simplified: \(\frac{-18}{12} \) simplifies to -1.5

Key concepts

Substituting Variables

Understanding how to substitute variables is a crucial skill in algebra that allows us to evaluate expressions based on specific conditions. In the given exercise, the expression \( \frac{x+y}{12} \) requires the substitution of the variables x and y with the values -24 and 6, respectively. This process involves replacing each variable with its given number, transforming the algebraic expression into a numerical one, which is easier to compute. For instance, the original expression turns into \( \frac{(-24)+6}{12} \), where the literal x is replaced with -24 and y with 6.

To perform the substitution accurately, make sure to use parentheses to maintain the integrity of the numbers, especially when they are negative. This will prevent mistakes during the following calculations, such as the simplification of the numerator, which is the next step in solving this problem.

Simplifying Numerators

Simplifying the numerator is the next step after substituting the variables. It is an essential process in handling algebraic fractions. In our example, the expression to simplify is \( \frac{(-24)+6}{12} \), where the numerator is -24 + 6. Simplification starts by performing the arithmetic operation in the numerator: -24 plus 6 equals -18, resulting in a new expression \( \frac{-18}{12} \).

When simplifying numerators, it's vital to:

• Perform calculations carefully, following the proper order of operations.
• Combine like terms if necessary, which typically involves addition and subtraction.
• Look out for opportunities to factor common elements, although in this case, we are only dealing with straightforward arithmetic.

In the context of our exercise, it becomes apparent that simplifying up to this point has brought us closer to the final answer, but the task is not finished until the fraction as a whole has been simplified.

Fraction Simplification

The final step in our exercise is fraction simplification, which is a fundamental concept in algebra. Once we have a simplified numerator, like our \( \frac{-18}{12} \) from the earlier step, our focus shifts to simplifying that fraction to its lowest terms. Simplification involves finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by that number.

In the given example, the GCD of 18 and 12 is 6. Dividing the numerator and denominator by 6, the fraction \( \frac{-18}{12} \) simplifies to \( \frac{-3}{2} \), which can also be written as -1.5 in decimal form. It's important to remember:

• Both the numerator and the denominator should be divided by the GCD to ensure the fraction's equivalence.
• The simplified form can be expressed as an improper fraction, a mixed number, or a decimal, depending on the context of the problem.
• If either the numerator or the denominator is a decimal, it may be converted to a fraction before further simplification.

Completing this process not only yields the simplest form of the fraction but also provides a more digestible interpretation of the value represented by the original algebraic expression.