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Chapter 8: Problem 5

\(x^{3}=2 y^{2} ; x=0, y=-2\)

Short Answer

Expert verified
Given values do not satisfy the equation because 0 鈮� 8.

Step by step solution

01

Understand the Given Equation

The equation given is: \[ x^3 = 2y^2 \]This is a relationship between the variables x and y.
02

Substitute the Given Values

Substitute the values of x and y into the equation. Here, x = 0 and y = -2.\[ (0)^3 = 2(-2)^2 \]
03

Simplify Both Sides of the Equation

Calculate each term separately:- On the left side: \(0^3 = 0\)- On the right side: \((-2)^2 = 4\), then multiply by 2: \(2 \times 4 = 8\)So, the equation simplifies to: \[ 0 = 8 \]
04

Check for Consistency

We find that the simplified equation is \( 0 = 8 \), which is false. Therefore, the given values do not satisfy the equation.

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Key Concepts

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

Variable Substitution
Variable substitution is a method used to solve equations by replacing variables with their known or given values.
This technique is essential as it helps simplify complex equations, making it easier to solve them.
Here is how it is applied in the given exercise:
Given the equation: \( x^3 = 2y^2 \)
We substitute the provided values: \( x = 0 \) and \( y = -2 \)
So the equation becomes: \( (0)^3 = 2(-2)^2 \)
This step transforms the abstract equation into a numerical form, which is the basis for solving it.
Equation Solving
Equation solving involves manipulating an equation to find the values of the variables that satisfy it.
In our example, after substituting the values, the equation is: \( (0)^3 = 2(-2)^2 \)
We need to simplify each side of the equation separately:
  • Left side: \( 0^3 = 0 \)
  • Right side: \( (-2)^2 = 4 \), then \( 2 \times 4 = 8 \)
The simplified equation is: \( 0 = 8 \)
Solving equations often requires these kinds of arithmetic operations and simplifications to reach a solution.
Consistency Check
A consistency check is used to verify if the obtained solution satisfies the original equation.
It ensures our calculations and substitutions maintain the truth of the equation.
Here, we have: \( 0 = 8 \)
This statement is false, indicating the substituted values \( x = 0 \) and \( y = -2 \) do not make the original equation true.
Therefore, upon performing a consistency check, we conclude the given values are inconsistent with the equation.
This step is crucial for validating the solution and ensuring the accuracy of our work.

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Most popular questions from this chapter

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\(y^{2}=x-1 ; x=3\)
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