Solve The Following System Of Equations:$\[ \begin{array}{l} -9x + 5y = -20 \\ 5x - 6y = -5 \end{array} \\]$\[ \begin{array}{l} x = \\ y = \end{array} \\]□

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Introduction

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and linear algebra. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables. We will use the method of substitution and elimination to find the solutions.

The System of Equations

The given system of equations is:

{ \begin{array}{l} -9x + 5y = -20 \\ 5x - 6y = -5 \end{array} \}

Step 1: Write Down the Equations

Let's write down the equations again:

{ \begin{array}{l} -9x + 5y = -20 \\ 5x - 6y = -5 \end{array} \}

Step 2: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to multiply the equations by necessary multiples. Let's multiply the first equation by 5 and the second equation by 9:

{ \begin{array}{l} -45x + 25y = -100 \\ 45x - 54y = -45 \end{array} \}

Step 3: Add the Equations

Now, let's add the two equations to eliminate the variable x:

{ \begin{array}{l} -45x + 25y = -100 \\ 45x - 54y = -45 \end{array} \}

Adding the two equations, we get:

{ -29y = -145 \}

Step 4: Solve for y

Now, let's solve for y:

{ -29y = -145 \}

Dividing both sides by -29, we get:

{ y = \frac{-145}{-29} \}

{ y = 5 \}

Step 5: Substitute the Value of y into One of the Original Equations

Now that we have the value of y, let's substitute it into one of the original equations to find the value of x. Let's use the first equation:

{ -9x + 5y = -20 \}

Substituting y = 5, we get:

{ -9x + 5(5) = -20 \}

{ -9x + 25 = -20 \}

Step 6: Solve for x

Now, let's solve for x:

{ -9x + 25 = -20 \}

Subtracting 25 from both sides, we get:

{ -9x = -45 \}

Dividing both sides by -9, we get:

{ x = \frac{-45}{-9} \}

{ x = 5 \}

Conclusion

In this article, we solved a system of two linear equations with two variables using the method of substitution and elimination. We found the values of x and y that satisfy the system of equations. The final answer is:

{ \begin{array}{l} x = 5 \\ y = 5 \end{array} \}

Discussion

Solving a system of linear equations is an essential concept in mathematics, particularly in algebra and linear algebra. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we used the method of substitution and elimination to find the solutions. The method of substitution involves substituting the value of one variable into one of the original equations to find the value of the other variable. The method of elimination involves adding or subtracting the equations to eliminate one of the variables.

Applications

Solving a system of linear equations has numerous applications in various fields, including:

  • Physics: Solving a system of linear equations is essential in physics to describe the motion of objects and to calculate the forces acting on them.
  • Engineering: Solving a system of linear equations is essential in engineering to design and analyze complex systems, such as electrical circuits and mechanical systems.
  • Computer Science: Solving a system of linear equations is essential in computer science to solve problems in computer graphics, game development, and machine learning.

Future Work

In the future, we can explore other methods for solving systems of linear equations, such as the method of matrices and the method of determinants. We can also apply the method of substitution and elimination to solve systems of linear equations with more than two variables.

References

  • [1]: "Linear Algebra and Its Applications" by Gilbert Strang
  • [2]: "Introduction to Linear Algebra" by Jim Hefferon
  • [3]: "Linear Algebra: A Modern Introduction" by David Poole

Note: The references provided are a selection of popular textbooks on linear algebra. There are many other resources available, including online courses, videos, and articles.

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Introduction

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and linear algebra. In our previous article, we solved a system of two linear equations with two variables using the method of substitution and elimination. In this article, we will answer some frequently asked questions (FAQs) about solving systems of linear equations.

Q: What is a system of linear equations?

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 combination of the variables, and the system is said to be consistent if it has a solution.

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

A system of linear equations has a solution if and only if the equations are consistent. This means that the system must have a unique solution, or the equations must be dependent (i.e., one equation is a multiple of the other).

Q: What are the methods for solving systems of linear equations?

There are several methods for solving systems of linear equations, including:

  • Method of Substitution: This method involves substituting the value of one variable into one of the original equations to find the value of the other variable.
  • Method of Elimination: This method involves adding or subtracting the equations to eliminate one of the variables.
  • Method of Matrices: This method involves representing the system of equations as a matrix and using matrix operations to solve the system.
  • Method of Determinants: This method involves using determinants to solve the system of equations.

Q: How do I choose the method for solving a system of linear equations?

The choice of method depends on the specific system of equations and the variables involved. In general, the method of substitution is useful for systems with two variables, while the method of elimination is useful for systems with more than two variables.

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

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

  • Not checking for consistency: Make sure that the system of equations is consistent before attempting to solve it.
  • Not using the correct method: Choose the correct method for solving the system of equations based on the variables involved.
  • Not checking for dependent equations: Make sure that the equations are not dependent before attempting to solve the system.

Q: How do I apply the method of substitution to solve a system of linear equations?

To apply the method of substitution, follow these steps:

  1. Choose one of the original equations: Select one of the original equations and solve for one of the variables.
  2. Substitute the value into the other equation: Substitute the value of the variable into the other equation to eliminate the variable.
  3. Solve for the remaining variable: Solve for the remaining variable using the resulting equation.

Q: How do I apply the method of elimination to solve a system of linear equations?

To apply the method of elimination, follow these steps:

  1. Choose two of the original equations: Select two of the original equations and add or subtract them to eliminate one of the variables.
  2. Solve for the remaining variable: Solve for the remaining variable using the resulting equation.
  3. Substitute the value into one of the original equations: Substitute the value of the variable into one of the original equations to find the value of the other variable.

Conclusion

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and linear algebra. In this article, we answered some frequently asked questions (FAQs) about solving systems of linear equations. We also provided a step-by-step guide on how to apply the method of substitution and elimination to solve systems of linear equations.

Discussion

Solving a system of linear equations has numerous applications in various fields, including physics, engineering, and computer science. In the future, we can explore other methods for solving systems of linear equations, such as the method of matrices and the method of determinants.

Future Work

In the future, we can apply the method of substitution and elimination to solve systems of linear equations with more than two variables. We can also explore other methods for solving systems of linear equations, such as the method of matrices and the method of determinants.

References

  • [1]: "Linear Algebra and Its Applications" by Gilbert Strang
  • [2]: "Introduction to Linear Algebra" by Jim Hefferon
  • [3]: "Linear Algebra: A Modern Introduction" by David Poole

Note: The references provided are a selection of popular textbooks on linear algebra. There are many other resources available, including online courses, videos, and articles.