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Chapter 1: Problem 6

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

Short Answer

Expert verified
The value of the expression is 1.6.

Step by step solution

01

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 \(-8 + 28 = 20\).
02

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 \(12.5\).
03

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 \(1.6\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Name the sets of numbers to which each number belongs. \(-4 . \overline{2}\)

Solve each inequality. Graph the solution set on a number line. $$ |4 k-8| < 20 $$

Solve each equation. Check your solutions. \(|x+5|=18\)

Mrs. Lucas earns a salary of \(\$ 34,000\) per year plus 1.5\(\%\) commission on her sales. If the average price of a car she sells is \(\$ 30,500\) , about how many cars must she sell to make an annual income of at least \(\$ 50,000 ?\) Write an inequality to describe this situation.

Solve each inequality. Then graph the solution set on a number line. \(12\left(\frac{1}{4}-\frac{n}{3}\right) \leq-6 n\)
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