Problem 56 - Algebra 2

Question

Evaluate the expression \(\frac{x}{x+y}\) for the given values of \(x\) and \(y\). \(x=2, y=8\)

Step-by-Step Solution

Answer

\( \frac{1}{5} \)

Step 1: Substitute the values

To evaluate the expression \( \frac{x}{x+y} \), first substitute the given values \( x=2 \) and \( y=8 \) into the expression. This gives you: \( \frac{2}{2 + 8} \).

Step 2: Simplify the expression

Now, simplify the denominator. Calculate \( 2 + 8 \) which equals \( 10 \). This modifies the expression to \( \frac{2}{10} \).

Step 3: Find the simplified fraction

To simplify \( \frac{2}{10} \) divide both the numerator and the denominator by their greatest common divisor, which is \( 2 \). You get \( \frac{1}{5} \).

Key Concepts

Substitution Method

The substitution method is a fundamental technique that simplifies expressions by replacing variables with known values. This method is an essential tool in algebra to evaluate expressions efficiently and accurately. For instance, if you have an expression like \( \frac{x}{x+y} \) and you're given \( x = 2 \) and \( y = 8 \), you can use substitution to directly replace \( x \) and \( y \) with these numbers. This transforms the original expression into \( \frac{2}{2+8} \). It’s often helpful to substitute variables underlining the given values. This method removes ambiguity, making the solving process clearer and more straightforward.

• Identify the expression you need to evaluate.
• Assign the given values to the proper variables in the expression.
• Proceed with any further calculations or simplifications.

Once substitution is done, the expression becomes easier to handle, allowing for seamless progression in solving.

Simplifying Fractions

Simplifying fractions is the process of making a fraction simpler or more concise, without altering its value. This process involves reducing the numerator and the denominator to their smallest possible integers. Taking our example \( \frac{2}{10} \), simplification is necessary to ensure we represent the fraction in its most basic form. First, calculate the sum \( 2 + 8 = 10 \) to simplify the expression to \( \frac{2}{10} \). The next step is to find a way to reduce this fraction. You do this by determining if there is a common factor between the numerator and the denominator that you can divide them both by, leading to the simplest version of the fraction.

• Identify the fraction.
• Check for common factors between the numerator and the denominator.
• Divide both by the greatest common factor to simplify.

Simplifying often makes fractions easier to use in calculations and compare with other values.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest number that evenly divides two or more numbers. Finding the GCD is particularly useful in simplifying fractions, as it helps reduce the numerator and the denominator to their smallest integers. In our exercise, the fraction \( \frac{2}{10} \) was simplified by first finding the GCD of 2 and 10, which is 2. This common divisor is then used to divide both the numerator and the denominator:

• Numerator: \( 2 \div 2 = 1 \)
• Denominator: \( 10 \div 2 = 5 \)

By understanding and using the GCD, you ensure that the fraction is presented in its simplest form, \( \frac{1}{5} \). This makes fractions easier to understand and use in further mathematical operations. Using the GCD not only simplifies fractions but enhances the understanding of number relationships and factorization.