Solve The Following System Of Equations:${ \begin{array}{l} 6x - 3y = -33 \ 5x + 2y = -23 \end{array} }$

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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 given system of equations as an example and provide a step-by-step guide on how to solve it.

The System of Equations

The given system of equations is:

{ \begin{array}{l} 6x - 3y = -33 \\ 5x + 2y = -23 \end{array} \}

Method 1: Substitution Method

One way to solve this system of equations is by using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.

Step 1: Solve the First Equation for x

We can start by solving the first equation for x:

6x - 3y = -33

6x = -33 + 3y

x = (-33 + 3y) / 6

Step 2: Substitute the Expression for x into the Second Equation

Now, we can substitute the expression for x into the second equation:

5x + 2y = -23

5((-33 + 3y) / 6) + 2y = -23

Step 3: Simplify the Equation

We can simplify the equation by multiplying both sides by 6:

5(-33 + 3y) + 12y = -138

-165 + 15y + 12y = -138

27y = 27

y = 1

Step 4: Find the Value of x

Now that we have the value of y, we can find the value of x by substituting y into one of the original equations:

x = (-33 + 3(1)) / 6

x = (-33 + 3) / 6

x = -30 / 6

x = -5

Method 2: Elimination Method

Another way to solve this system of equations is by using the elimination method. This method involves adding or subtracting the equations to eliminate one variable.

Step 1: Multiply the Equations by Necessary Multiples

We can start by multiplying the equations by necessary multiples to make the coefficients of y's in both equations the same:

6x - 3y = -33

5x + 2y = -23

6x - 3y = -33

-2(5x + 2y) = -2(-23)

-10x - 4y = 46

Step 2: Add the Equations

Now, we can add the equations to eliminate the variable y:

6x - 3y = -33

-10x - 4y = 46

-4x - 7y = 13

Step 3: Solve for x

We can solve for x by isolating it on one side of the equation:

-4x = 13 + 7y

x = (-13 - 7y) / 4

Step 4: Find the Value of y

Now that we have the value of x, we can find the value of y by substituting x into one of the original equations:

6x - 3y = -33

6((-13 - 7y) / 4) - 3y = -33

Step 5: Simplify the Equation

We can simplify the equation by multiplying both sides by 4:

6(-13 - 7y) - 12y = -132

-78 - 42y - 12y = -132

-54y = 54

y = -1

Step 6: Find the Value of x

Now that we have the value of y, we can find the value of x by substituting y into one of the original equations:

x = (-13 - 7(-1)) / 4

x = (-13 + 7) / 4

x = -6 / 4

x = -3/2

Conclusion

In this article, we have solved a system of two linear equations with two variables using the substitution method and the elimination method. We have shown that both methods can be used to solve the system of equations and have provided step-by-step guides on how to do so. We have also provided examples of how to solve the system of equations using both methods.

Final Answer

The final answer is:

x = -5

y = 1

Note: The final answer is the same for both methods.

References

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

Future Work

In the future, we can explore other methods for solving systems of linear equations, such as the graphing method and the matrix method. We can also explore more complex systems of linear equations, such as systems with three or more variables.

Limitations

One limitation of this article is that it only provides examples of how to solve systems of linear equations with two variables. In the future, we can provide examples of how to solve systems of linear equations with three or more variables.

Conclusion

In conclusion, solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and linear algebra. We have shown that both the substitution method and the elimination method can be used to solve a system of two linear equations with two variables. We have also provided step-by-step guides on how to do so and have provided examples of how to solve the system of equations using both methods.

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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: A system of linear equations has a solution if and only if the two equations are consistent, meaning that they do not contradict each other. If the two equations are inconsistent, then the system has no solution.

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

A: The two main methods for solving systems of linear equations are the substitution method and the elimination method.

Q: What is the substitution method?

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

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one variable.

Q: How do I choose which method to use?

A: You can choose which method to use based on the coefficients of the variables in the two equations. If the coefficients of one variable are the same in both equations, then the elimination method is a good choice. If the coefficients of one variable are different in both equations, then the substitution method is a good choice.

Q: What if I have a system of linear equations with three or more variables?

A: If you have a system of linear equations with three or more variables, then you can use the same methods as before, but you will need to use more equations to eliminate the variables.

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 if I have a system of linear equations with no solution?

A: If you have a system of linear equations with no solution, then the two equations are inconsistent, meaning that they contradict each other.

Q: What if I have a system of linear equations with infinitely many solutions?

A: If you have a system of linear equations with infinitely many solutions, then the two equations are dependent, meaning that one equation is a multiple of the other.

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. You can graph the two equations on the same coordinate plane and find the point of intersection, which is the solution to the system.

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 before solving the system
  • Not using the correct method for solving the system
  • Not checking if the solution is a valid solution
  • Not checking if the solution is unique

Q: How can I practice solving systems of linear equations?

A: You can practice solving systems of linear equations by working through examples and exercises in a textbook or online resource. You can also try solving systems of linear equations on your own, using a calculator or graphing calculator to check your work.

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 describe 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, computer vision, and machine learning.