Problem 1 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The value of the expression is \(\frac{7}{4}\).

Step 1: Substitute the Given Values

Start by substituting the given values of the variables into the expression. The expression is \( \frac{x+5}{x+2} \). We substitute \( x = 2 \). This gives:\[\frac{2+5}{2+2}\]

Step 2: Perform the Addition in the Numerator and Denominator

Calculate the addition operations in both the numerator and the denominator. - Numerator: \(2 + 5 = 7\)- Denominator: \(2 + 2 = 4\)This modifies the expression to:\[\frac{7}{4}\]

Step 3: Simplify the Fraction

Since \(7/4\) is already in its simplest form (with no common factors other than 1), we do not need any further simplification. The value of the expression \(\frac{x+5}{x+2}\) when \(x = 2\) is:\[\frac{7}{4}\]

Key Concepts

Substitution

Substitution is a fundamental technique used in algebra to simplify expressions and solve equations. It involves replacing variables in an expression with their specific values. In this exercise, we substitute \(x = 2\) into the expression \(\frac{x+5}{x+2}\). This involves directly inserting the number 2 wherever the variable \(x\) appears.

• The numerator, originally \(x + 5\), becomes \(2 + 5\).
• The denominator, \(x + 2\), transforms into \(2 + 2\).

By performing substitution, we convert an abstract algebraic expression into a concrete arithmetic one, making it easier to handle. Once substitution is completed, the expression is prepared for further arithmetic operations.

Fractions

Fractions represent a division between two quantities. They consist of a numerator and a denominator. Understanding fractions is crucial when dealing with algebraic expressions, especially when substituting values. Here, we are working with the fraction \(\frac{7}{4}\), derived from the substitution and arithmetic done in the previous steps.

• Numerator: Represents the top part of the fraction. In this case, it is 7, obtained by adding \(2+5\).
• Denominator: Represents the bottom part of the fraction, which is 4 after adding \(2+2\).

Fractions are used not only in algebra but in many areas of mathematics to express parts of a whole. Being comfortable with manipulating fractions allows for greater flexibility in solving equations and understanding mathematical concepts.

Simplification

Simplification involves reducing an expression to its simplest form. In this exercise, after substituting and performing arithmetic operations, we obtained the fraction \(\frac{7}{4}\).

• First, check if the fraction can be simplified. This involves checking for common factors between the numerator and denominator.
• Since 7 is a prime number and does not share any factors with 4 other than 1, the fraction is already in its simplest form.

Thus, no additional simplification is necessary. Simplification is a key part of problem-solving in algebra as it helps in obtaining the most concise and exact form of an expression, making it easier to interpret and use.