Problem 4 - Algebra A Combined Function

Question

Find the value of the following expressions when \(x=2, y=-2,\) and \(z=-5\). $$ \frac{z}{z^{2}-5} $$

Step-by-Step Solution

Answer

The value of the expression is \(-\frac{1}{4}\).

Step 1: Substitute given values into the expression

We have the expression \( \frac{z}{z^2 - 5} \). Substitute the given value for \( z = -5 \) into the expression. This yields the expression \( \frac{-5}{(-5)^2 - 5} \).

Step 2: Simplify the denominator

Calculate \((-5)^2\) which equals 25. Now subtract 5 from 25: \(25 - 5 = 20\). The simplified denominator is 20.

Step 3: Evaluate the entire expression

Now that we have the expression \( \frac{-5}{20} \), simplify it. Divide both the numerator and denominator by 5 to get \( \frac{-1}{4} \).

Key Concepts

Substitution Method

The substitution method is a crucial step in evaluating algebraic expressions. It involves replacing the variables in an expression with their given numerical values. This makes the expression much simpler to handle.

For example, considering our original expression \( \frac{z}{z^2 - 5} \), we substitute the value of \( z=-5 \). This means wherever you see \( z \) in the expression, you replace it with \(-5\). So, the expression becomes \( \frac{-5}{(-5)^2 - 5} \).

The purpose of substitution is to transform the variable expression into a numerical one, allowing you to perform arithmetic operations without variables. It is essential to ensure accuracy during substitution by carefully identifying all instances of a variable in the expression.

Numerator and Denominator Simplification

After substituting the values into an expression, the next step is to simplify both the numerator and the denominator.

Let's explore this with our example \( \frac{-5}{(-5)^2 - 5} \). The numerator, which is \(-5\), is already a simple number, so no further simplification is needed there.

Moving on to the denominator, you need to calculate \((-5)^2\), which involves squaring \(-5\). Squaring any negative number makes it positive because a negative times a negative is positive. Therefore, \((-5)^2=25\).

Next, subtract the 5 from 25:

• \(25 - 5 = 20\)

The denominator is now simplified to 20.

Proper simplification of the numerator and the denominator is essential, as it sets the stage for evaluating the expression.

Expression Evaluation

Expression evaluation is the final step where you arrive at a single, simplified value for your expression. After substitution and simplification, you evaluate the expression by performing the arithmetic operations.

For instance, with our simplified expression \( \frac{-5}{20} \), the next step is to divide the numerator by the denominator.

Dividing \(-5\) by 20 gives:

• \( \frac{-5}{20} = \frac{-1}{4} \)

So, the evaluated result of the expression is \( \frac{-1}{4} \).

• Be careful with signs: Since \(-5\) is negative, the result is also negative.
• Final answers should typically be reduced to their simplest form, ensuring clarity and correctness.

Understanding how to evaluate expressions accurately is key to mastering algebra, as it confirms your ability to manipulate and solve equations effectively.