Problem 66 - Algebra A Combined Function

Question

Evaluate each expression when \(x=12, y=8,\) and \(z=4\). $$ \frac{y^{2}+x}{x^{2}+3 y} $$

Step-by-Step Solution

Answer

The value of the expression is \( \frac{19}{42} \).

Step 1: Substitute the given values

Replace the variables in the expression with the given numbers. You have: \( y = 8 \), \( x = 12 \), and \( z = 4 \). Substitute these into the expression: \( \frac{y^{2}+x}{x^{2}+3y} = \frac{8^{2}+12}{12^{2}+3 \times 8} \).

Step 2: Calculate the numerator

Calculate the value of the numerator by evaluating \( 8^{2} + 12 \). This is \( 64 + 12 = 76 \).

Step 3: Calculate the denominator

Calculate the value of the denominator by evaluating \( 12^{2} + 3 \times 8 \). This is \( 144 + 24 = 168 \).

Step 4: Simplify the fraction

Now that we have \( \frac{76}{168} \), simplify this fraction by finding the greatest common divisor of 76 and 168, which is 4. Thus, \( \frac{76}{168} = \frac{76 \div 4}{168 \div 4} = \frac{19}{42} \).

Key Concepts

Substitution Method

The substitution method is a common technique in algebra where you replace variables in an expression or equation with their given values to simplify or solve it. This method is extremely useful when dealing with complex expressions or equations that involve multiple variables. In our task, we have the expression \( \frac{y^{2}+x}{x^{2}+3y} \), and we know that \( x=12 \), \( y=8 \), and \( z=4 \). To evaluate the expression, we substitute these values for the variables:

• Replace \( y \) with 8
• Replace \( x \) with 12
• Ignore \( z \) since it's not part of the expression

Thus, the expression becomes \( \frac{8^2 + 12}{12^2 + 3 \times 8} \). This substitution simplifies the evaluation process, turning an expression with variables into one with only numbers. This step sets the foundation for the next parts of the solution process.

Fraction Simplification

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no other common divisors except 1. After substituting the known values into the expression, we evaluated both the numerator and the denominator separately:

• The numerator is \( 8^2 + 12 = 76 \)
• The denominator is \( 12^2 + 3 \times 8 = 168 \)

Now we have the fraction \( \frac{76}{168} \). Both 76 and 168 can be divided by their greatest common divisor (GCD), which is 4. Dividing both by 4, we get:

• \( 76 \div 4 = 19 \)
• \( 168 \div 4 = 42 \)

Thus, the simplified fraction is \( \frac{19}{42} \). Simplifying fractions is crucial in mathematics as it often makes it easier to see relationships and comparisons, and is generally preferred in final answers.

Numerical Evaluation

Once the substitution is done and the fraction is simplified, the numerical evaluation gives us the final numeric value of an expression. This practice is used to convert an algebraic expression into a specific number, which is especially useful in practical applications where exact numbers are necessary.For this expression, after substituting the values and simplifying the fraction, we arrive at \( \frac{19}{42} \). Sometimes, you might need to convert this to a decimal or percentage for different contexts, like reporting your results in a science project or determining probabilities.In our example, converting \( \frac{19}{42} \) to a decimal would involve dividing 19 by 42, which results in approximately 0.452. Having these different representations of numbers can be extremely useful, depending on the requirement of the exercise or real-world application.