Solve The System Of Equations:${ \begin{array}{l} 9x + 9y = 11 \ -9x - 9y = -9 \ \end{array} }$

Introduction

Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a system of two linear equations with two variables. We will use the given system of equations as an example and provide a step-by-step solution.

The System of Equations

The given system of equations is:

$$\begin{array}{l} 9x + 9y = 11 \\ -9x - 9y = -9 \\ \end{array}$$

Understanding the System

Before we start solving the system, let's understand what we are dealing with. We have two linear equations with two variables, x and y. The first equation is:

$$9x + 9y = 11$$

This equation represents a line in the coordinate plane. The second equation is:

$$-9x - 9y = -9$$

This equation also represents a line in the coordinate plane. Our goal is to find the values of x and y that satisfy both equations simultaneously.

Method 1: Addition and Subtraction

One way to solve this system is to use the method of addition and subtraction. We can add the two equations together to eliminate one of the variables.

$$\begin{array}{l} (9x + 9y) + (-9x - 9y) = 11 + (-9) \\ 0x + 0y = 2 \\ \end{array}$$

However, this results in a contradiction, as the left-hand side of the equation is equal to 0, while the right-hand side is equal to 2. This means that the system has no solution.

Method 2: Multiplication and Addition

Another way to solve this system is to multiply one of the equations by a constant and then add the two equations together.

Let's multiply the first equation by 1 and the second equation by 1.

$$\begin{array}{l} 9x + 9y = 11 \\ -9x - 9y = -9 \\ \end{array}$$

Now, let's add the two equations together.

$$\begin{array}{l} (9x + 9y) + (-9x - 9y) = 11 + (-9) \\ 0x + 0y = 2 \\ \end{array}$$

However, this results in a contradiction, as the left-hand side of the equation is equal to 0, while the right-hand side is equal to 2. This means that the system has no solution.

Method 3: Substitution

Another way to solve this system is to use the method of substitution. We can solve one of the equations for one of the variables and then substitute that expression into the other equation.

Let's solve the first equation for x.

$$9x + 9y = 11$$

$$9x = 11 - 9y$$

$$x = \frac{11 - 9y}{9}$$

Now, let's substitute this expression for x into the second equation.

$$-9x - 9y = -9$$

$$-9\left(\frac{11 - 9y}{9}\right) - 9y = -9$$

$$-11 + 9y - 9y = -9$$

$$-11 = -9$$

However, this results in a contradiction, as the left-hand side of the equation is equal to -11, while the right-hand side is equal to -9. This means that the system has no solution.

Method 4: Elimination

Another way to solve this system is to use the method of elimination. We can multiply one of the equations by a constant and then add the two equations together to eliminate one of the variables.

Let's multiply the first equation by 1 and the second equation by 1.

$$\begin{array}{l} 9x + 9y = 11 \\ -9x - 9y = -9 \\ \end{array}$$

Now, let's add the two equations together.

$$\begin{array}{l} (9x + 9y) + (-9x - 9y) = 11 + (-9) \\ 0x + 0y = 2 \\ \end{array}$$

However, this results in a contradiction, as the left-hand side of the equation is equal to 0, while the right-hand side is equal to 2. This means that the system has no solution.

Conclusion

In this article, we have seen four different methods for solving a system of two linear equations with two variables. However, in each case, we have found that the system has no solution. This is because the two equations are inconsistent, meaning that they cannot be true at the same time.

Final Answer

The final answer is that the system of equations has no solution.

References

• [1] "Systems of Linear Equations" by Math Open Reference
• [2] "Solving Systems of Linear Equations" by Khan Academy

Note

Introduction

In our previous article, we discussed the concept of solving systems of linear equations and provided a step-by-step guide on how to solve a system of two linear equations with two variables. However, we found that the system had no solution. In this article, we will answer some frequently asked questions (FAQs) related to solving systems 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 two or more variables. Each equation is a linear equation, meaning that it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

Q: How do I know if a system of linear equations has a solution?

A: To determine if a system of linear equations has a solution, you need to check if the two equations are consistent or inconsistent. If the two equations are consistent, then the system has a solution. If the two equations are inconsistent, then the system has no solution.

Q: What is the difference between a consistent and an inconsistent system?

A: A consistent system is a system of linear equations that has a solution. An inconsistent system is a system of linear equations that has no solution.

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

A: There are several methods for solving a system of linear equations, including:

• Addition and subtraction
• Multiplication and addition
• Substitution
• Elimination

Q: What is the addition and subtraction method?

A: The addition and subtraction method involves adding or subtracting the two equations to eliminate one of the variables.

Q: What is the multiplication and addition method?

A: The multiplication and addition method involves multiplying one of the equations by a constant and then adding the two equations to eliminate one of the variables.

Q: What is the substitution method?

A: The substitution method involves solving one of the equations for one of the variables and then substituting that expression into the other equation.

Q: What is the elimination method?

A: The elimination method involves multiplying one of the equations by a constant and then adding the two equations to eliminate one of the variables.

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

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

• Not checking if the two equations are consistent or inconsistent
• Not using the correct method for solving the system
• Not following the steps of the method correctly
• Not checking the solution for consistency

Q: How do I check if a solution is consistent?

A: To check if a solution is consistent, you need to plug the values of the variables into both equations and check if the equations are true.

Q: What are some real-world applications of solving systems of linear equations?

A: Solving systems of linear equations has many real-world applications, including:

• Physics: Solving systems of linear equations is used to model the motion of objects in physics.
• Engineering: Solving systems of linear equations is used to design and optimize systems in engineering.
• Economics: Solving systems of linear equations is used to model economic systems and make predictions about economic trends.
• Computer Science: Solving systems of linear equations is used in computer science to solve problems in computer graphics, game development, and machine learning.

Conclusion

In this article, we have answered some frequently asked questions related to solving systems of linear equations. We have discussed the concept of a system of linear equations, how to determine if a system has a solution, and how to solve a system using different methods. We have also discussed some common mistakes to avoid when solving systems of linear equations and how to check if a solution is consistent.