Problem 55 - Algebra 2

Question

Evaluate the expression \(\frac{x}{x+y}\) for the given values of \(x\) and \(y\). \(x=4, y=4\)

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{1}{2}\).

Step 1: Substitute the Given Values

Take the given expression \(\frac{x}{x+y}\) and replace \(x\) and \(y\) with the provided values. Hence, substitute \(x=4\) and \(y=4\) into the expression which becomes \(\frac{4}{4+4}\).

Step 2: Simplify the Denominator

Calculate the sum in the denominator: \(4 + 4 = 8\). So the expression becomes \(\frac{4}{8}\).

Step 3: Simplify the Fraction

Simplify \(\frac{4}{8}\) by dividing both numerator and denominator by their greatest common divisor, which is 4. Therefore: \(\frac{4}{8} = \frac{4 \div 4}{8 \div 4} = \frac{1}{2}\).

Key Concepts

Substitution

When evaluating expressions, substitution is the key to replacing variables with specific values. In our case, the expression is \(\frac{x}{x+y}\). Since we are given \(x=4\) and \(y=4\), we substitute these values into the expression. This step is crucial because it transforms a general expression into a numerical one that can be easily solved. For example:

• Take the expression \(\frac{x}{x+y}\),
• Substitute \(x = 4\) and \(y = 4\),
• This results in \(\frac{4}{4+4}\).

After substitution, evaluate any operations in the expression to proceed to the next steps. By substituting, we move from an abstract idea to actual numbers we can compute.

Simplifying Fractions

Simplifying fractions is essential to make expressions easier to understand and work with. Once we have substituted the values \(x = 4\) and \(y = 4\) into \(\frac{x}{x+y}\), the expression becomes \(\frac{4}{8}\).
This fraction can be simplified. Simplification involves reducing the fraction to its simplest form. This is done by dividing both the numerator and the denominator by their greatest common divisor. For this expression:

• The numerator is 4.
• The denominator is 8.

Both 4 and 8 can be evenly divided by 4, the greatest common divisor. Thus, \(\frac{4}{8}\) simplifies to \(\frac{1}{2}\).
The simplified form \(\frac{1}{2}\) retains the same value as the original fraction but is tidier and easier to understand.

Greatest Common Divisor

The greatest common divisor (GCD) is an important concept in math, especially when simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder. Identifying the GCD is crucial in reducing fractions to their simplest form. For our example,

• The numbers involved are 4 (numerator) and 8 (denominator).
• The common divisors of 4 and 8 are 1, 2, and 4.
• The greatest of these divisors is 4.

Therefore, the GCD of 4 and 8 is 4, allowing us to simplify \(\frac{4}{8}\) to \(\frac{1}{2}\). Finding the GCD makes it much easier to handle fractions, keep calculations clean, and results in expressions that are simple to interpret and work with.