Problem 4 - Algebra 2

Question

Evaluate each expression if \(x=4, y=-2,\) and \(z=3.5\) \(\frac{x^{2}-y}{z+2.5}\)

Step-by-Step Solution

Answer

The value of the expression is 3.

Step 1: Substitute Variables into the Expression

First, we substitute the given values of the variables into the expression. We have that \(x = 4\), \(y = -2\), and \(z = 3.5\). The expression is \(\frac{x^{2} - y}{z + 2.5}\). Substituting these values, we get \(\frac{4^{2} - (-2)}{3.5 + 2.5}\).

Step 2: Simplify Numerator

Next, let's simplify the numerator \(4^{2} - (-2)\). Calculating \(4^{2}\), we get 16. Then, subtracting \(-2\) is the same as adding 2, resulting in \(16 + 2 = 18\).

Step 3: Simplify Denominator

Now, simplify the denominator \(3.5 + 2.5\). By adding these two numbers, we get 6.

Step 4: Evaluate the Expression

Now that we have the simplified expression \(\frac{18}{6}\), we can divide 18 by 6 to get the result. Dividing, we obtain 3.

Key Concepts

Substitution Method

In the world of algebra, the substitution method is an essential skill. It involves replacing variables in a given expression with specific values. This process allows for the evaluation of the expression using known numbers rather than unknowns.
Let's consider the problem: evaluate \(\frac{x^{2}-y}{z+2.5}\) given that \(x=4\), \(y=-2\), and \(z=3.5\).
- Replace \(x\) with 4, \(y\) with -2, and \(z\) with 3.5.
- After substitution, the expression becomes \(\frac{4^{2}-(-2)}{3.5+2.5}\).
By systematically substituting these values, we convert the expression from a general form into something we can easily evaluate. This is the power of the substitution method: making complex expressions manageable.

Simplification

Simplification is about making algebraic expressions easier to understand and work with. When an expression is complex or too lengthy, simplification is the process of reducing it to its simplest form, often by combining like terms or performing basic arithmetic operations.
In our example:

• Start by simplifying the numerator: First calculate \(4^{2}\), which is 16. Next, simplify \(-(-2)\), which is the same as adding 2.
• This results in the numerator being 16 + 2 = 18.
• For the denominator, simply add \(3.5 + 2.5\) to get 6.

The goal of simplification here is to make the expression \(\frac{4^{2} - (-2)}{3.5 + 2.5}\) much more concise and easier to evaluate, leading directly to \(\frac{18}{6}\).

Numerical Evaluation

Numerical evaluation is the act of calculating a precise numerical value from a simplified expression. After substitution and simplification, the next step is to compute the actual value.
In this exercise, once we achieve \(\frac{18}{6}\), we proceed by performing the division:

• 18 divided by 6 equals 3.

This step is crucial as it transforms a theoretical expression into a numerical answer, allowing us to understand the practical meaning of the expression.

Basic Algebra Operations

Basic algebra operations including addition, subtraction, multiplication, and division underlie many mathematical processes. Mastery of these is fundamental for tackling algebraic expressions.
Consider our problem:

• Simplify the expression by recognizing subtracting a negative number as an addition. \(-(-2)\) becomes +2, simplifying the process.
• Multiply and add as needed, here \(4^{2}=16\).
• Division, the final operation, \(18/6=3\), gives the result.

These basic operations might seem straightforward, but they are critical in handling more complex algebraic manipulations.
Understanding each operation thoroughly ensures accurate evaluation and prepares students for advanced mathematical challenges.