Problem 54 - Algebra 2

Question

Evaluate the expression \(\frac{x}{x+y}\) for the given values of \(x\) and \(y\). \(x=3, y=2\)

Step-by-Step Solution

Answer

The value of the expression is \( \frac{3}{5} \).

Step 1: Substitute the given values into the expression

Start by replacing the variables in the expression with the given values. The expression is \( \frac{x}{x+y} \). Substitute \( x = 3 \) and \( y = 2 \) into the expression, resulting in \( \frac{3}{3+2} \).

Step 2: Simplify the denominator

Add the numbers in the denominator. In this case, add \(3\) and \(2\) which equals \(5\). Therefore, the expression simplifies to \( \frac{3}{5} \).

Step 3: Final evaluation

The expression \( \frac{3}{5} \) can be evaluated as a fraction. There are no further simplifications possible, so the final result is \( \frac{3}{5} \).

Key Concepts

Substitution

Substitution is a fundamental concept in algebra that involves replacing variables in an expression with given values. This is often the first step in evaluating an algebraic expression. Imagine substitution as a simple swap: you take out the letter or symbol representing something and replace it with a number. For instance, in the expression \(\frac{x}{x+y}\), both \(x\) and \(y\) are placeholders for numbers. Once you substitute \(x = 3\) and \(y = 2\), the expression becomes \(\frac{3}{3+2}\).

It's crucial to execute substitution correctly to ensure accurate evaluation. Always double-check that you've replaced the right variable with the correct value. Treat the variables like puzzle pieces, where each value completes the picture of the expression.

Simplification

Simplification involves reducing an expression into its most basic form without changing its value. This step helps to make an expression easier to understand and solve.

When simplifying, you might need to perform operations like addition, subtraction, multiplication, or division. In our fraction example, once \(x\) and \(y\) are replaced by 3 and 2, respectively, the expression \(\frac{3}{3+2}\) has the denominator \(3+2\). Simplifying the sum, \(3+2\) gives us 5. As such, the expression becomes \(\frac{3}{5}\).

• Identify and execute basic arithmetic operations.
• Always follow the order of operations, sometimes remembered by PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Check if the expression can be reduced further.

Though the expression \(\frac{3}{5}\) cannot be simplified further, ensuring you reduce where possible is essential in finding the simplest form.

Fraction Evaluation

Fraction evaluation is the final step where you interpret the result of an expression presented as a fraction. It's important to understand what the fraction represents and how it fits into the bigger picture of problem-solving.

A fraction is essentially a division of two numbers: the numerator (top number) tells you how many parts you have, while the denominator (bottom number) tells you the total parts considered. In the expression \(\frac{3}{5}\), 3 is the numerator, and 5 is the denominator. This tells us we have 3 out of 5 parts.

• A fraction simplifies our understanding of parts of a whole.
• The simplest form of a fraction often provides the clearest understanding.

In this case, \(\frac{3}{5}\) is already in its simplest form, so this is the final result of the algebraic expression. Evaluating fractions correctly helps in understanding proportional relationships in diverse real-world contexts.