Problem 5 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The expression evaluates to 24.

Step 1: Substitute the given values

First, substitute the given values for each variable in the expression. Replace \(x\) with 4, \(y\) with -2, and \(z\) with 3.5 in the expression \(\frac{x+2y^2}{x-z}\).

Step 2: Calculate the expression in the numerator

The numerator of the expression is \(x+2y^2\). By substituting the values, it becomes \(4 + 2(-2)^2\). Calculate \((-2)^2 = 4\) and then multiply it by 2 to get 8. The expression in the numerator simplifies to \(4 + 8 = 12\).

Step 3: Calculate the expression in the denominator

The denominator of the expression is \(x-z\). Substituting the values, it becomes \(4 - 3.5\). This simplifies directly to \(0.5\).

Step 4: Evaluate the expression

Now, divide the simplified numerator by the simplified denominator: \(\frac{12}{0.5}\). Dividing 12 by 0.5 is equivalent to multiplying 12 by 2, resulting in \(24\).

Key Concepts

Substitution in Algebra

Substitution in algebra is a helpful method to solve for unknown variables by replacing them with known values. In the given exercise, we're tasked with evaluating an expression by substituting specific values for the variables \(x, y,\) and \(z\). When substituting, it’s important to:

• Identify each variable in the expression and its given value.
• Replace each variable with its corresponding value accurately.
• Ensure that each substitution step is clearly followed to avoid mistakes.

In this example, we substitute \(x=4\), \(y=-2\), and \(z=3.5\) directly into the expression \(\frac{x+2y^2}{x-z}\). This process transforms an algebraic expression into a numerical one, making it simpler to manage.

Order of Operations

Once substitution is completed, the next essential step is using the "order of operations" to solve the expression correctly. This rule dictates the sequence in which operations should be performed to avoid ambiguity. We remember this sequence with the acronym PEMDAS, standing for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our exercise, the order of operations is critical. After substituting, we have \(4 + 2(-2)^2\). Here, \((-2)^2\) must be calculated first, giving us \(4\). Then, multiply by 2 to get \(8\). Finally, add \(4\) for a total of \(12\). Similarly, for the denominator \(4 - 3.5\), simple subtraction suffices. Following these rules ensures that each component of the expression is computed correctly before proceeding.

Simplifying Fractions

After evaluating the numerator and the denominator, the final step is simplifying the fraction. Simplifying makes it easier to interpret and understand results by expressing them in their simplest form. For the expression \(\frac{12}{0.5}\), simplification involves dividing the numerator by the denominator. In general, simplifying involves:

• Identifying the greatest common factor (GCF) if possible, to reduce the fraction's components.
• Performing the division accurately.
• Converting division into an equivalent multiplication by using the reciprocal, if needed.

In our case, dividing \(12\) by \(0.5\) is equal to multiplying \(12\) by \(2\), leading to a final result of \(24\). Through simplifying fractions, we achieve a more straightforward answer for the evaluated expression.