Solve The System Of Equations.$ \begin{array}{r} 3y - 9x = -6 \ 4y - 12x = -8 \end{array} }$Select The Correct Answer Below And, If Necessary, Fill In The Answer A. The Solution Is { \square$ $ (Type An Ordered Pair.)B. There Are

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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 means 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.

The System of Equations

The given system of equations is:

{ \begin{array}{r} 3y - 9x = -6 \\ 4y - 12x = -8 \end{array} \}

Method of Elimination

One of the methods to solve a system of linear equations is the method of elimination. This method involves adding or subtracting the equations in the system to eliminate one of the variables. In this case, we can multiply the first equation by 4 and the second equation by 3 to make the coefficients of y's in both equations the same.

Step 1: Multiply the First Equation by 4

Multiplying the first equation by 4 gives us:

12y−36x=−2412y - 36x = -24

Step 2: Multiply the Second Equation by 3

Multiplying the second equation by 3 gives us:

12y−36x=−2412y - 36x = -24

Step 3: Subtract the Second Equation from the First Equation

Subtracting the second equation from the first equation gives us:

0=00 = 0

This means that the two equations are equivalent, and we can eliminate one of the variables by subtracting one equation from the other.

Step 4: Solve for x

Since the two equations are equivalent, we can solve for x by substituting the value of y from one equation into the other equation. Let's substitute the value of y from the first equation into the second equation:

4y−12x=−84y - 12x = -8

4(3x+2)−12x=−84(3x + 2) - 12x = -8

12x+8−12x=−812x + 8 - 12x = -8

8=−88 = -8

This is a contradiction, which means that the system of equations has no solution.

Conclusion

In conclusion, the system of equations ${ \begin{array}{r} 3y - 9x = -6 \ 4y - 12x = -8 \end{array} }$ has no solution. This is because the two equations are equivalent, and we can eliminate one of the variables by subtracting one equation from the other. The resulting equation is a contradiction, which means that the system of equations has no solution.

Answer

The correct answer is:

B. There are no solutions.

Note: There is no ordered pair that satisfies both equations in the system.

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Introduction

In our previous article, we discussed how to solve a system of linear equations using the method of elimination. We also saw that the given system of equations had no solution. 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: 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 means finding the values of the variables that satisfy all the equations in the system.

Q: What are the different methods to solve a system of linear equations?

A: There are several methods to solve a system of linear equations, including:

  • Method of Elimination: This method involves adding or subtracting the equations in the system to eliminate one of the variables.
  • Method of Substitution: This method involves solving one equation for one variable and then substituting that expression into the other equation.
  • Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the difference between a dependent and an independent system of equations?

A: A dependent system of equations is a system in which the equations are equivalent, and there are an infinite number of solutions. A independent system of equations is a system in which the equations are not equivalent, and there is a unique solution.

Q: How do I know if a system of equations has a solution or not?

A: To determine if a system of equations has a solution or not, you can use the following steps:

  1. Check if the equations are equivalent: If the equations are equivalent, then the system has no solution.
  2. Check if the equations are inconsistent: If the equations are inconsistent, then the system has no solution.
  3. Check if the equations are consistent: If the equations are consistent, then the system has a solution.

Q: What is the importance of solving systems of linear equations?

A: Solving systems of linear equations is an important skill in mathematics and has many real-world applications, including:

  • Physics and Engineering: Solving systems of linear equations is used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
  • Computer Science: Solving systems of linear equations is used in computer graphics, game development, and machine learning.
  • Economics: Solving systems of linear equations is used to model economic systems and make predictions about the behavior of markets.

Conclusion

In conclusion, solving systems of linear equations is an important skill in mathematics that has many real-world applications. By understanding the different methods to solve systems of linear equations and being able to identify whether a system has a solution or not, you can apply this skill to a wide range of problems.

Frequently Asked Questions

Q: What is the difference between a system of linear equations and a system of nonlinear equations?

A: A system of linear equations is a set of linear equations that involve the same set of variables. A system of nonlinear equations is a set of nonlinear equations that involve the same set of variables.

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

A: Solving a system of nonlinear equations is more complex than solving a system of linear equations. There are several methods to solve a system of nonlinear equations, including:

  • Numerical Methods: These methods involve using numerical techniques, such as the Newton-Raphson method, to find an approximate solution.
  • Analytical Methods: These methods involve using analytical techniques, such as substitution and elimination, to find an exact solution.

Q: What is the importance of solving systems of nonlinear equations?

A: Solving systems of nonlinear equations is an important skill in mathematics and has many real-world applications, including:

  • Physics and Engineering: Solving systems of nonlinear equations is used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
  • Computer Science: Solving systems of nonlinear equations is used in computer graphics, game development, and machine learning.
  • Economics: Solving systems of nonlinear equations is used to model economic systems and make predictions about the behavior of markets.