Problem 57 - Algebra 2

Question

Evaluate the expression \(\frac{x}{x+y}\) for the given values of \(x\) and \(y\). \(x=5, y=10\)

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{1}{3}\).

Step 1: Substitute Values

Replace the given values of \(x\) and \(y\) into the expression. Here, substitute \(x = 5\) and \(y = 10\) into the expression \(\frac{x}{x+y}\). This gives us: \(\frac{5}{5 + 10}\).

Step 2: Simplify the Denominator

Add the values in the denominator. So, calculate \(5 + 10\), which equals 15. The expression now becomes \(\frac{5}{15}\).

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{5}{15}\) by finding the greatest common divisor of 5 and 15, which is 5. Divide the numerator and the denominator by 5: \(\frac{5 \div 5}{15 \div 5} = \frac{1}{3}\).

Key Concepts

Substitution

Substitution is a fundamental step in solving algebraic expressions, especially when specific values are provided for the variables involved. In the given expression, \(\frac{x}{x+y}\), we have values specified for the variables: \(x = 5\) and \(y = 10\). Substitution involves replacing these variables with the provided numerical values.

• Identify the variables in the algebraic expression.
• Replace each variable with the corresponding value.
• Re-evaluate the expression using arithmetic operations.

In this example, substituting \(x = 5\) and \(y = 10\) into \(\frac{x}{x+y}\) transforms it into \(\frac{5}{5+10}\). This step is essential because it converts the problem into a numeric equation, which can then be simplified further.
Substitution is a straightforward but crucial tool when working with algebra, and it helps bridge the transition from variables into concrete numbers.

Simplification

Simplification in mathematics involves reducing an equation or expression to its most basic form. After substitution,
the expression \(\frac{5}{5+10}\) appears. Here, simplification embarks on breaking down or solving complex operations into simple arithmetic,
like addition, subtraction, multiplication, or division, to reach the simplest form.

• Add or subtract numbers if necessary. In this case, solve \(5 + 10\) to get 15.
• Rewrite the expression as \(\frac{5}{15}\) following the arithmetic operation.

Now, the expression is more straightforward and leads us into the next key stage: simplifying fractions. Simplification makes it easier to interpret and solve problems, especially when dealing with complex algebraic structures.
Practicing simplification gives one a robust understanding of how numbers and operations work together.

Fractions

Fractions represent a part of a whole. In algebraic expressions, simplifying fractions to their lowest terms makes calculation and interpretation easier.
When faced with a fraction like \(\frac{5}{15}\), we try to reduce it by finding a common divisor for both the numerator and the denominator.

• Find the greatest common divisor (GCD) for both numerator and denominator. In \(5\) and \(15\), the GCD is \(5\).
• Divide both the numerator and the denominator by the GCD.
• The simplified form of \(\frac{5}{15}\) is \(\frac{1}{3}\) after dividing by \(5\), as \(\frac{5 \div 5}{15 \div 5} = \frac{1}{3}\).

Simplifying fractions is crucial as it leads to easier computation and comparison of values.
It also enhances the clarity of algebraic expressions, enabling a better understanding of mathematical relationships and operations.