Dividing the Second Equation by the First Equation

Oct 13, 2025 by windailys.com 142 views

Solving the System of Equations: Finding the Values of a and b

In this article, we will delve into the world of mathematics and explore a system of equations involving two unknown variables, a and b. The system consists of two equations, each with a unique relationship between a and b. Our goal is to find the values of a and b that satisfy both equations simultaneously. We will analyze the given equations, apply mathematical techniques, and arrive at the correct solution.

The system of equations is given as:

{ \begin{array}{l} 32 = a b^2 \\ 128 = a b^3 \end{array} \}

We are tasked with finding the values of a and b that satisfy both equations. To begin, let's examine the first equation: $32 = a b^2$ . This equation implies that the product of a and $b^2$ is equal to 32.

Dividing the Second Equation by the First Equation

To simplify the system of equations, we can divide the second equation by the first equation. This will help us eliminate one of the variables and solve for the other.

{ \frac{128}{32} = \frac{a b^3}{a b^2} \}

Simplifying the left-hand side, we get:

{ 4 = b \}

Now, we have found the value of b, which is 4.

Substituting the Value of b into the First Equation

We can substitute the value of b into the first equation to solve for a.

{ 32 = a (4)^2 \}

Simplifying the right-hand side, we get:

{ 32 = a (16) \}

Dividing both sides by 16, we get:

{ a = 2 \}

In conclusion, we have found the values of a and b that satisfy the system of equations. The correct solution is:

{ a = 2; b = 4 \}

This solution satisfies both equations, and we have arrived at the correct answer through mathematical analysis and simplification.

The system of equations presented in this article is a classic example of a mathematical problem that requires careful analysis and simplification. By dividing the second equation by the first equation, we were able to eliminate one of the variables and solve for the other. This technique is commonly used in mathematics to simplify complex systems of equations.

In this article, we have demonstrated the importance of mathematical techniques in solving systems of equations. We have also shown that with careful analysis and simplification, we can arrive at the correct solution to a complex mathematical problem.

The final answer is:

{ a = 2; b = 4 \}

In our previous article, we explored a system of equations involving two unknown variables, a and b. We analyzed the given equations, applied mathematical techniques, and arrived at the correct solution. In this article, we will provide a Q&A section to further clarify the concepts and provide additional insights.

Q: What is the main difference between the two equations in the system?

A: The main difference between the two equations is the power of b. In the first equation, b is squared, while in the second equation, b is cubed. This difference in power affects the relationship between a and b.

Q: Why did we divide the second equation by the first equation?

A: We divided the second equation by the first equation to eliminate one of the variables and solve for the other. This technique is commonly used in mathematics to simplify complex systems of equations.

Q: What is the significance of the value of b being 4?

A: The value of b being 4 is significant because it allows us to substitute this value into the first equation and solve for a. This substitution is a crucial step in finding the correct solution.

Q: Can we use other mathematical techniques to solve this system of equations?

A: Yes, there are other mathematical techniques that can be used to solve this system of equations. However, the technique of dividing the second equation by the first equation is a straightforward and efficient method.

Q: What is the relationship between a and b in this system of equations?

A: The relationship between a and b in this system of equations is that a is proportional to b. Specifically, a is equal to 2 times b squared.

Q: Can we generalize this solution to other systems of equations?

A: While this solution is specific to this particular system of equations, the mathematical techniques used can be applied to other systems of equations. However, the specific solution will depend on the coefficients and variables involved.

Q: What is the importance of solving systems of equations in mathematics?

A: Solving systems of equations is an essential part of mathematics, as it allows us to model real-world problems and make predictions. In this article, we have demonstrated the importance of mathematical techniques in solving systems of equations.

In conclusion, we have provided a Q&A section to further clarify the concepts and provide additional insights into solving the system of equations. We have demonstrated the importance of mathematical techniques in solving systems of equations and highlighted the significance of the value of b being 4.

The final answer is:

{ a = 2; b = 4 \}

This solution satisfies both equations, and we have arrived at the correct answer through mathematical analysis and simplification.

For further reading and practice, we recommend the following resources:

Khan Academy: Systems of Equations
MIT OpenCourseWare: Linear Algebra
Wolfram Alpha: Systems of Equations

These resources provide additional information and practice problems to help you master the concepts of solving systems of equations.