Problem 56 - Algebra 1: Concepts and Skills

Question

Evaluate the expression when x 3 and y 6. $$ \frac{3 x}{x+y} $$

Step-by-Step Solution

Answer

The evaluated expression is 1.

Step 1: Substitute the given values

We replace x with 3 and y with 6 in the expression, giving us \(\frac{3 \cdot 3}{3+6}\)

Step 2: Evaluate the numerator and denominator

We perform the multiplication in the numerator, and the addition in the denominator: \(\frac{9}{9}\)

Step 3: Divide

We divide the numerator by the denominator, resulting in the final value of 1

Key Concepts

Substitution

Substitution is the process of replacing variables in expressions with their corresponding values. In our exercise, we were given specific values for the variables \(x = 3\) and \(y = 6\). These values are substituted into the expression \(\frac{3x}{x+y}\).
The substitution step transforms the expression into \(\frac{3 \cdot 3}{3 + 6}\). This is an essential step before any calculation begins, as it simplifies the expression into a form that involves only numbers, making it easier to evaluate.
Substitution often acts as the foundation for solving more complex expressions, ensuring that each variable is replaced accurately according to the problem's requirements.

Numerator and Denominator

These terms are crucial when working with fractions. In any fraction \(\frac{a}{b}\), 'a' is the numerator, and 'b' is the denominator. In our expression, after substitution, our numerator became \(3 \cdot 3\), and the denominator became \(3 + 6\).
To evaluate the expression, we first handle the operations in both parts separately:

• Numerator: Multiply \(3 \times 3\) to get \(9\)
• Denominator: Add \(3 + 6\) to get \(9\)

This careful evaluation ensures accuracy, as simplifying each part individually helps maintain clarity in the calculations.
Understanding how to work with numerators and denominators is vital for mastering fractions and expressions in mathematics.

Division

Division is the last, but crucial step in evaluating our expression. Once you've simplified both the numerator and the denominator, the next step is to divide the two. In our example, we ended up with the fraction \(\frac{9}{9}\).
Dividing 9 by 9 simplifies to 1, as any number divided by itself equals 1. This concept is fundamental to fractions and ensures that you get the simplest form of the expression.
Being comfortable with division and understanding how it reduces fractions helps solve many mathematical problems, ensuring accuracy and simplicity in your final answers.