Solve The System Of Equations Below.$\[ \begin{aligned} -3x + 6y &= 9 \\ 5x + 7y &= -49 \end{aligned} \\]A. \[$(1, -2)\$\] B. \[$\left(-2, \frac{1}{2}\right)\$\] C. \[$(-2, -7)\$\] D. \[$(-7, -2)\$\]

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Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. 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 solution.

The System of Equations

The system of equations we will be solving is:

{ \begin{aligned} -3x + 6y &= 9 \\ 5x + 7y &= -49 \end{aligned} \}

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to multiply the equations by necessary multiples such that the coefficients of either x or y will be the same in both equations. Let's multiply the first equation by 5 and the second equation by 3.

{ \begin{aligned} -15x + 30y &= 45 \\ 15x + 21y &= -147 \end{aligned} \}

Step 2: Add the Equations

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

{ -15x + 30y + 15x + 21y = 45 - 147 \}

Simplifying the equation, we get:

{ 51y = -102 \}

Step 3: Solve for y

Now, let's solve for y by dividing both sides of the equation by 51.

{ y = \frac{-102}{51} \}

Simplifying the equation, we get:

{ y = -2 \}

Step 4: 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 solve for x. Let's use the first equation.

{ -3x + 6(-2) = 9 \}

Simplifying the equation, we get:

{ -3x - 12 = 9 \}

Adding 12 to both sides of the equation, we get:

{ -3x = 21 \}

Dividing both sides of the equation by -3, we get:

{ x = -7 \}

Conclusion

Therefore, the solution to the system of equations is x = -7 and y = -2.

Answer

The correct answer is:

{ (-7, -2) \}

Discussion

This problem is a great example of how to solve a system of linear equations using the method of substitution and elimination. By multiplying the equations by necessary multiples and adding them, we were able to eliminate one of the variables and solve for the other. This method can be used to solve systems of linear equations with two or more variables.

Tips and Tricks

  • When solving a system of linear equations, it's essential to identify the variables and the constants in each equation.
  • Use the method of substitution and elimination to eliminate one of the variables and solve for the other.
  • Make sure to multiply the equations by necessary multiples to eliminate the variable.
  • Add the equations to eliminate the variable and solve for the other variable.

Related Topics

  • Solving systems of linear equations using the method of substitution
  • Solving systems of linear equations using the method of elimination
  • Solving systems of linear equations with three or more variables

References

  • [1] "Solving Systems of Linear Equations" by Math Open Reference
  • [2] "Solving Systems of Linear Equations" by Khan Academy
  • [3] "Solving Systems of Linear Equations" by MIT OpenCourseWare
    Solving a System of Linear Equations: Q&A =====================================

Introduction

In our previous article, we discussed how to solve a system of linear equations using the method of substitution and elimination. 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 are solved simultaneously to find the values of the variables.

Q: How do I know which method to use to solve a system of linear equations?

You can use either the method of substitution or elimination to solve a system of linear equations. The method you choose will depend on the coefficients of the variables in the equations.

Q: What is the method of substitution?

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

Q: What is the method of elimination?

The method of elimination involves adding or subtracting the equations to eliminate one of the variables.

Q: How do I multiply the equations by necessary multiples?

To multiply the equations by necessary multiples, you need to multiply both sides of each equation by a number that will make the coefficients of either x or y the same in both equations.

Q: How do I add the equations to eliminate the variable?

To add the equations, you need to add both sides of each equation together.

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

If you have a system of linear equations with three or more variables, you can use the method of substitution or elimination to solve for two of the variables, and then use the values of those variables to solve for the third variable.

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

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 will be the solution to the system.

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

If you have a system of linear equations with no solution, it means that the two equations are inconsistent and there is no point that satisfies both equations.

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

If you have a system of linear equations with infinitely many solutions, it means that the two equations are dependent and there are an infinite number of points that satisfy both equations.

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

Yes, you can use a computer program to solve a system of linear equations. There are many computer programs available that can solve systems of linear equations, including MATLAB, Python, and R.

Conclusion

Solving systems of linear equations is an essential skill in mathematics and is used in many real-world applications. By understanding the methods of substitution and elimination, you can solve systems of linear equations with ease.

Tips and Tricks

  • Make sure to identify the variables and the constants in each equation.
  • Use the method of substitution or elimination to solve the system of linear equations.
  • Multiply the equations by necessary multiples to eliminate the variable.
  • Add the equations to eliminate the variable and solve for the other variable.
  • Use a graphing calculator or computer program to solve the system of linear equations.

Related Topics

  • Solving systems of linear equations using the method of substitution
  • Solving systems of linear equations using the method of elimination
  • Solving systems of linear equations with three or more variables

References

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