Which Is The Best Approximate Solution Of The System Of Linear Equations $y = 1.5x - 1$ And $y = 1$?A. (0.33, 1) B. (1.33, 1) C. (1.83, 1) D. (2.33, 1)

Feb 27, 2025 by ADMIN 155 views

**Solving a System of Linear Equations: A Step-by-Step Approach**

Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on finding the best approximate solution of the system of linear equations given by $y = 1.5x - 1$ and $y = 1$ .

Understanding the Problem

The problem requires us to find the approximate solution of the system of linear equations $y = 1.5x - 1$ and $y = 1$ . To solve this problem, we need to find the values of $x$ and $y$ that satisfy both equations.

Step 1: Write Down the Equations

The two equations are:

$y = 1.5x - 1$
$y = 1$

Step 2: Equate the Two Equations

Since both equations are equal to $y$ , we can equate them to each other:

$1.5x - 1 = 1$

Step 3: Solve for x

To solve for $x$ , we need to isolate $x$ on one side of the equation. We can do this by adding $1$ to both sides of the equation:

$1.5x = 2$

Next, we can divide both sides of the equation by $1.5$ to solve for $x$ :

$x = \frac{2}{1.5}$

$x = \frac{20}{15}$

$x = \frac{4}{3}$

$x = 1.33$

Step 4: Find the Value of y

Now that we have found the value of $x$ , we can substitute it into one of the original equations to find the value of $y$ . We will use the first equation:

$y = 1.5x - 1$

$y = 1.5(1.33) - 1$

$y = 1.995 - 1$

$y = 0.995$

However, we are given that $y = 1$ . This means that our solution is not exact, but rather an approximation.

Comparing the Solutions

We are given four possible solutions:

A. (0.33, 1) B. (1.33, 1) C. (1.83, 1) D. (2.33, 1)

We have found that $x = 1.33$ and $y = 0.995$ . However, we are given that $y = 1$ . This means that our solution is not exact, but rather an approximation.

Conclusion

In conclusion, the best approximate solution of the system of linear equations $y = 1.5x - 1$ and $y = 1$ is (1.33, 1).

Final Answer

The final answer is B. (1.33, 1).

Additional Information

It's worth noting that the solution (1.33, 1) is an approximation, and the exact solution may be different. However, based on the given information, (1.33, 1) is the best approximate solution.

References

[1] Linear Algebra and Its Applications, 4th Edition, by Gilbert Strang
[2] Introduction to Linear Algebra, 5th Edition, by Gilbert Strang

Keywords

System of linear equations
Approximate solution
Linear algebra
Mathematics
Frequently Asked Questions (FAQs) About Solving a System of Linear Equations ================================================================================

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you can use the following steps:

Write down the equations
Equate the two equations
Solve for one variable
Substitute the value of the variable into one of the original equations to find the value of the other variable

Q: What is the difference between an exact solution and an approximate solution?

A: An exact solution is a solution that satisfies all the equations in the system exactly, while an approximate solution is a solution that satisfies all the equations in the system approximately.

Q: How do I know if my solution is exact or approximate?

A: You can check if your solution is exact or approximate by plugging the values of the variables into the original equations and checking if they satisfy the equations exactly or approximately.

Q: What is the best way to solve a system of linear equations?

A: The best way to solve a system of linear equations is to use a method such as substitution or elimination. These methods involve solving for one variable and then substituting the value of the variable into one of the original equations to find the value of the other variable.

Q: Can I use a calculator to solve a system of linear equations?

A: Yes, you can use a calculator to solve a system of linear equations. Many calculators have built-in functions for solving systems of linear equations.

Q: What are some common mistakes to avoid when solving a system of linear equations?

A: Some common mistakes to avoid when solving a system of linear equations include:

Not checking if the solution satisfies all the equations in the system
Not using the correct method for solving the system (such as substitution or elimination)
Not plugging the values of the variables into the original equations to check if they satisfy the equations exactly or approximately

Q: How do I know if my solution is unique or not?

A: You can check if your solution is unique or not by plugging the values of the variables into the original equations and checking if they satisfy the equations exactly or approximately. If the solution satisfies all the equations in the system exactly, then it is a unique solution. If the solution satisfies all the equations in the system approximately, then it is not a unique solution.

Q: What is the importance of solving a system of linear equations?

A: Solving a system of linear equations is important because it allows you to find the values of the variables that satisfy all the equations in the system. This can be useful in a variety of applications, such as physics, engineering, and economics.

Q: Can I use a computer program to solve a system of linear equations?

A: Yes, you can use a computer program to solve a system of linear equations. Many computer programs, such as MATLAB and Python, have built-in functions for solving systems of linear equations.

Q: What are some common applications of solving a system of linear equations?

A: Some common applications of solving a system of linear equations include:

Physics: Solving systems of linear equations is used to model the motion of objects and to solve problems involving forces and energies.
Engineering: Solving systems of linear equations is used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Solving systems of linear equations is used to model economic systems and to solve problems involving supply and demand.

Q: How do I know if my solution is valid or not?

A: You can check if your solution is valid or not by plugging the values of the variables into the original equations and checking if they satisfy the equations exactly or approximately. If the solution satisfies all the equations in the system exactly, then it is a valid solution. If the solution satisfies all the equations in the system approximately, then it is not a valid solution.

Q: What are some common errors to avoid when solving a system of linear equations?

A: Some common errors to avoid when solving a system of linear equations include:

Not checking if the solution satisfies all the equations in the system
Not using the correct method for solving the system (such as substitution or elimination)
Not plugging the values of the variables into the original equations to check if they satisfy the equations exactly or approximately

Q: Can I use a graphing calculator to solve a system of linear equations?

A: Yes, you can use a graphing calculator to solve a system of linear equations. Many graphing calculators have built-in functions for solving systems of linear equations.

Q: What are some common applications of graphing calculators in solving systems of linear equations?

A: Some common applications of graphing calculators in solving systems of linear equations include:

Visualizing the solution space
Finding the intersection points of two or more lines
Solving systems of linear equations with multiple variables

Q: How do I know if my solution is a unique solution or not?

A: You can check if your solution is a unique solution or not by plugging the values of the variables into the original equations and checking if they satisfy the equations exactly or approximately. If the solution satisfies all the equations in the system exactly, then it is a unique solution. If the solution satisfies all the equations in the system approximately, then it is not a unique solution.

Q: What are some common mistakes to avoid when using a graphing calculator to solve a system of linear equations?

A: Some common mistakes to avoid when using a graphing calculator to solve a system of linear equations include:

Not checking if the solution satisfies all the equations in the system
Not using the correct method for solving the system (such as substitution or elimination)
Not plugging the values of the variables into the original equations to check if they satisfy the equations exactly or approximately