Find The Solution Of This System Of Equations. Separate The \[$x\$\]- And \[$y\$\]-values With A Comma.$\[ \begin{array}{c} x + 12y = -26 \\ x - Y = 0 \end{array} \\]

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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 solving a system of two linear equations with two variables, x and y.

The System of Equations

The given system of equations is:

x+12y=−26x−y=0\begin{array}{c} x + 12y = -26 \\ x - y = 0 \end{array}

Our goal is to find the values of x and y that satisfy both equations.

Method 1: Substitution Method

One way to solve this system of equations is by using the substitution method. We can solve one of the equations for one variable and then substitute that expression into the other equation.

Let's solve the second equation for x:

x=yx = y

Now, substitute this expression for x into the first equation:

y+12y=−26y + 12y = -26

Combine like terms:

13y=−2613y = -26

Divide both sides by 13:

y=−2y = -2

Now that we have found the value of y, we can substitute it back into the expression for x:

x=−2x = -2

Method 2: Elimination Method

Another way to solve this system of equations is by using the elimination method. We can multiply both equations by necessary multiples such that the coefficients of y's in both equations are the same:

Multiply the second equation by 12:

12x−12y=012x - 12y = 0

Now, add both equations to eliminate the y-variable:

x+12y+12x−12y=−26+0x + 12y + 12x - 12y = -26 + 0

Combine like terms:

13x=−2613x = -26

Divide both sides by 13:

x=−2x = -2

Now that we have found the value of x, we can substitute it back into one of the original equations to find the value of y:

x−y=0x - y = 0

Substitute x = -2:

−2−y=0-2 - y = 0

Add 2 to both sides:

−y=2-y = 2

Multiply both sides by -1:

y=−2y = -2

Conclusion

In this article, we have solved a system of two linear equations with two variables, x and y. We have used two different methods, the substitution method and the elimination method, to find the values of x and y that satisfy both equations. The values of x and y are x = -2 and y = -2.

Separating the x- and y-values

The solution to the system of equations is x = -2, y = -2. We can separate the x- and y-values with a comma:

x = -2, y = -2

Final Answer

Introduction

In our previous article, we solved a system of two linear equations with two variables, x and y. In this article, we will answer some frequently asked questions 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 the same set of variables. For example:

x+12y=−26x−y=0\begin{array}{c} x + 12y = -26 \\ x - y = 0 \end{array}

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

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

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

The two main methods for solving a system of linear equations are the substitution method and the elimination method.

Q: What is the substitution method?

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

Let's solve the second equation for x:

x=yx = y

Now, substitute this expression for x into the first equation:

y+12y=−26y + 12y = -26

Q: What is the elimination method?

The elimination method involves multiplying both equations by necessary multiples such that the coefficients of y's in both equations are the same, and then adding both equations to eliminate the y-variable. For example:

Multiply the second equation by 12:

12x−12y=012x - 12y = 0

Now, add both equations to eliminate the y-variable:

x+12y+12x−12y=−26+0x + 12y + 12x - 12y = -26 + 0

Q: How do I know which method to use?

The choice of method depends on the coefficients of the variables in the two equations. If the coefficients of one variable are the same in both equations, use the elimination method. If the coefficients of one variable are different in both equations, use the substitution method.

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

If you have a system of three or more linear equations, you can use the same methods as before, but you may need to use additional techniques, such as the use of matrices or the method of elimination with multiple variables.

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

Yes, you can use a calculator to solve a system of linear equations. Most graphing calculators and computer algebra systems have built-in functions for solving systems of linear equations.

Conclusion

In this article, we have answered some frequently asked questions about solving systems of linear equations. We have discussed the two main methods for solving a system of linear equations, the substitution method and the elimination method, and we have provided examples of how to use each method.

Final Tips

  • Make sure to check your work by plugging the solution back into both equations.
  • Use a calculator or computer algebra system to check your work if you are unsure.
  • Practice solving systems of linear equations to become more comfortable with the methods and techniques.

Common Mistakes to Avoid

  • Not checking your work by plugging the solution back into both equations.
  • Not using the correct method for the given system of equations.
  • Not simplifying the equations before solving them.

Additional Resources

  • Online resources, such as Khan Academy and Mathway, offer video lessons and interactive exercises for solving systems of linear equations.
  • Textbooks and online resources, such as Wolfram Alpha, provide detailed explanations and examples of how to solve systems of linear equations.

Final Answer

The final answer is that solving a system of linear equations requires a clear understanding of the two main methods, the substitution method and the elimination method, and the ability to apply these methods to a variety of problems.