Problem 6 - Algebra 2

Question

Evaluate each expression if \(x=4, y=-2,\) and \(z=3.5\) \(\frac{y^{3}+2 x z}{x^{2}-z}\)

Step-by-Step Solution

Answer

The value of the expression is 1.6.

Step 1: Calculate the numerator

To find the value of the numerator, substitute the given values into the equation, which is \(y^{3} + 2xz\). Substituting \(y = -2\), \(x = 4\), and \(z = 3.5\) into the expression gives \((-2)^{3} + 2(4)(3.5)\). Calculate \((-2)^{3} = -8\) and \(2(4)(3.5) = 28\). The numerator becomes \(-8 + 28 = 20\).

Step 2: Calculate the denominator

To find the value of the denominator, substitute the given values into the equation, which is \(x^{2} - z\). Substitute \(x = 4\) and \(z = 3.5\). Calculate \(4^{2} = 16\) and then \(16 - 3.5 = 12.5\). The denominator is \(12.5\).

Step 3: Divide the numerator by the denominator

Combine the calculated numerator and denominator by dividing. So, we need to find \(\frac{20}{12.5}\). Perform the division to get \(1.6\).

Key Concepts

Substitution

Substitution is one of the key steps in solving algebraic expressions. It involves replacing variables with their specific values to simplify the expression. In the context of the given exercise, substitution was necessary to evaluate the expression for specific values of \(x\), \(y\), and \(z\).When you are asked to evaluate an expression like \(\frac{y^{3}+2 x z}{x^{2}-z}\), you simply replace each variable with the given numbers:

• Replace \(x\) with 4
• Replace \(y\) with -2
• Replace \(z\) with 3.5

Thus, substitution transforms the expression into numbers that can be further simplified into an answer. Properly executing this step simplifies the entire problem-solving process and reduces potential errors.

Numerator and Denominator

Understanding the terms numerator and denominator is crucial for solving fraction problems. The numerator is the part of a fraction above the line, representing the number of parts we have. The denominator is below the line, indicating the total number of parts the whole is divided into.In our exercise, we evaluated the expression \(y^{3} + 2xz\) to find the numerator. After substituting the values, we calculated:

• \((-2)^{3} = -8\)
• \(2 \times 4 \times 3.5 = 28\)
• The numerator becomes \(-8 + 28 = 20\)

Next, we solved for the denominator using \(x^{2} - z\):

• \(4^{2} = 16\)
• \(16 - 3.5 = 12.5\)

With a clear separation and understanding of numerator and denominator, one can accurately perform the division to find the final answer.

Division

Division is the mathematical operation used to find out how many times one number is contained within another. In the context of fractions, it determines the 'value' or 'portion' of the division of the numerator by the denominator.In the exercise's final step, we divide the calculated numerator by the denominator:

• We found the numerator to be 20.
• The denominator was calculated to be 12.5.
• Thus, we perform the division \(\frac{20}{12.5}\).

By carrying out the division, we find that \(\frac{20}{12.5} = 1.6\). Division requires accuracy: set up the numbers correctly and ensure calculations are precise. Understanding each division component allows smoother problem-solving and prevents common mistakes.