Problem 62 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\(\frac{19}{42}\)

Step 1: Substitute the Values

Start by substituting the given values for \(x\), \(y\), and \(z\) into the expression: \(\frac{y^{2}+x}{x^{2}+3y}\). Given, \(x=12\), \(y=8\), and \(z=4\), we substitute: \(\frac{8^{2}+12}{12^{2}+3 \times 8}\).

Step 2: Calculate the Numerator

Calculate the expression in the numerator: \(8^{2} + 12\). This simplifies to \(64 + 12 = 76\).

Step 3: Calculate the Denominator

Calculate the expression in the denominator: \(12^{2} + 3 \times 8\). This simplifies to \(144 + 24 = 168\).

Step 4: Divide the Numerator by Denominator

Now divide the calculated numerator by the denominator: \(\frac{76}{168}\). Simplify this fraction: both numerator and denominator can be divided by their greatest common divisor which is 4, resulting in \(\frac{19}{42}\).

Key Concepts

Fractions Simplification

Simplifying fractions is an essential skill in algebra, as it makes expressions easier to understand and compare. To simplify a fraction, the goal is to divide the numerator and the denominator by their greatest common divisor (GCD). This process reduces the fraction without changing its value.

For example, in the expression \(\frac{76}{168}\), we simplify it by finding the GCD of 76 and 168, which is 4. Dividing both the numerator and the denominator by 4 gives us \(\frac{19}{42}\). Now, the fraction is in its simplest form.

Remember:

• Always look for the highest number that divides both numbers evenly.
• Keep the value of the fraction the same by performing this operation simultaneously on the numerator and denominator.

Substitution Method

The substitution method is a handy tool for evaluating algebraic expressions. It involves replacing variables with specific numbers to simplify and solve equations. Particularly in exercises where values for variables are given, like \(x=12\), \(y=8\), and \(z=4\), substitution helps to turn a general expression into a numerical one.

To use this method, replace each variable in the expression with its corresponding value. For the expression \(\frac{y^{2}+x}{x^{2}+3y}\), substitute \(y=8\) and \(x=12\):

- Substitute \(y\) with 8, so \(y^2\) becomes \(8^2\).
- Substitute \(x\) with 12, thus the whole expression tranforms into \(\frac{8^2+12}{12^2+3\times8}\).

This method breaks down complex expressions into simpler arithmetical steps.

Numerator and Denominator Calculation

Computing the numerator and the denominator separately is a crucial step when evaluating and simplifying algebraic fractions. By doing so, each part of the fraction becomes a manageable calculation, making the entire process clearer.

For example:

• Numerator: Start by calculating \(8^2 + 12\), which results in \(76\).
• Denominator: Calculate \(12^2 + 3 \times 8\), resulting in \(168\).

Performing these calculations individually makes it easier to plug them back into the fraction \(\frac{76}{168}\).

Always ensure your arithmetic is correct for both parts before you attempt to simplify the fraction. This prevents errors and ensures your final answer is accurate.

Greatest Common Divisor (GCD)

Finding the greatest common divisor (GCD) is important when simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder. The GCD helps to reduce fractions to their simplest form, making them easier to work with.

To find the GCD:

• List all factors of each number.
• Identify the largest factor that appears in both lists.

For instance, to simplify \(\frac{76}{168}\), determine the GCD of 76 and 168. The factors of 76 are 1, 2, 4, 19, 38, and 76, while those of 168 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 56, 84, and 168. The greatest common factor is 4.

Divide both 76 and 168 by 4 to achieve the simplified fraction \(\frac{19}{42}\). Understanding and finding the GCD efficiently is vital in manipulating and simplifying fractions.