Problem 2 - Algebra A Combined Function

Question

Find the value of the following expressions when \(x=2, y=-2,\) and \(z=-5\). $$ \frac{x+8}{x+1} $$

Step-by-Step Solution

Answer

The value is \(\frac{10}{3}\).

Step 1: Substitute the Values

Replace each variable in the expression with the given values: substitute \(x = 2\) into the expression.

Step 2: Evaluate the Numerator

Start by evaluating the numerator, which is \(x + 8\). So, replace \(x\) with 2 to get \(2 + 8 = 10\).

Step 3: Evaluate the Denominator

Next, evaluate the denominator, which is \(x + 1\). Replace \(x\) with 2 to get \(2 + 1 = 3\).

Step 4: Simplify the Expression

Now divide the evaluated numerator by the evaluated denominator: \(\frac{10}{3}\).

Key Concepts

Variable Substitution

Variable substitution is a fundamental concept in algebra that involves replacing variables in an expression with given numerical values. In our given expression \( \frac{x+8}{x+1} \), we have the variables \(x\), \(y\), and \(z\). To solve the expression, we only substitute \(x\) as its value is relevant here.- Given: \(x = 2\)- Substitute 2 in place of \(x\) in the expression.By substituting \(x = 2\), the expression becomes \( \frac{2+8}{2+1} \). This step is crucial as it sets the stage for further simplification of the expression by converting it into a numerically evaluable form.It's like replacing ingredients in a recipe with their actual measurements before you start cooking. Each piece needs to be prepared before the final dish is ready.

Numerator and Denominator

When dealing with fractions, understanding the numerator and denominator is vital. In the fraction \( \frac{2+8}{2+1} \),- The **numerator** is the value on top (\(2 + 8\))- The **denominator** is the value on the bottom (\(2 + 1\))The numerator represents how many parts you have, while the denominator indicates how many equal parts the whole is divided into. Think of a cake cut into slices: the numerator tells you how many slices you have, and the denominator tells you how many slices make up the whole cake.First, we evaluate the numerator: \(2 + 8 = 10\). Then, we evaluate the denominator: \(2 + 1 = 3\). Now, the entire fraction reads \( \frac{10}{3} \), which allows us to proceed to simplifying if needed.

Expression Simplification

Simplifying an expression means reducing it to its simplest form to make it easier to understand or calculate. In algebra, this usually involves performing arithmetic or algebraic operations.Once we've substituted variables and evaluated both the numerator and the denominator, we arrive at the fraction \( \frac{10}{3} \). This expression is already simplified, meaning there are no further calculations or reductions possible. Simplification often helps in making calculations easier and in verifying results. However, in some cases, further simplification might involve converting the fraction into decimals or mixed numbers, but only if required. In our context, \( \frac{10}{3} \) is an acceptable and simplified result of the original expression.