What Is The Solution { (a, C)$}$ To This System Of Linear Equations?${ \begin{array}{r} 2a - 3c = -6 \ a + 2c = 11 \end{array} }$A. { \left(-\frac{76}{7}, \frac{17}{7}\right)$}$B. { (3, -4)$} C . \[ C. \[ C . \[ (3,

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Introduction to Systems of Linear Equations

Systems of linear equations are a fundamental concept in mathematics, particularly in algebra and geometry. They consist of two or more linear equations that are solved simultaneously to find the values of the variables involved. In this article, we will focus on solving a system of two linear equations with two variables, a and c. We will use the method of substitution and elimination to find the solution to the system.

The System of Linear Equations

The given system of linear equations is:

{ \begin{array}{r} 2a - 3c = -6 \\ a + 2c = 11 \end{array} \}

Step 1: Multiply the Two Equations by Necessary Multiples

To eliminate one of the variables, we need to multiply the two equations by necessary multiples such that the coefficients of either a or c are the same in both equations. Let's multiply the first equation by 1 and the second equation by 2.

{ \begin{array}{r} 2a - 3c = -6 \\ 2(a + 2c) = 2(11) \end{array} \}

This simplifies to:

{ \begin{array}{r} 2a - 3c = -6 \\ 2a + 4c = 22 \end{array} \}

Step 2: Subtract the First Equation from the Second Equation

Now, let's subtract the first equation from the second equation to eliminate the variable a.

{ (2a + 4c) - (2a - 3c) = 22 - (-6) \}

This simplifies to:

{ 7c = 28 \}

Step 3: Solve for c

Now, let's solve for c by dividing both sides of the equation by 7.

{ c = \frac{28}{7} \}

This simplifies to:

{ c = 4 \}

Step 4: Substitute the Value of c into One of the Original Equations

Now that we have the value of c, let's substitute it into one of the original equations to solve for a. We will use the second equation.

{ a + 2c = 11 \}

Substituting c = 4, we get:

{ a + 2(4) = 11 \}

This simplifies to:

{ a + 8 = 11 \}

Step 5: Solve for a

Now, let's solve for a by subtracting 8 from both sides of the equation.

{ a = 11 - 8 \}

This simplifies to:

{ a = 3 \}

Conclusion

In conclusion, the solution to the system of linear equations is a = 3 and c = 4. Therefore, the correct answer is:

B. (3, -4)

This solution can be verified by plugging the values of a and c back into the original equations to ensure that they are true.

Discussion

The solution to the system of linear equations can be found using various methods, including substitution and elimination. In this article, we used the elimination method to find the solution. The elimination method involves multiplying the two equations by necessary multiples such that the coefficients of either a or c are the same in both equations, and then subtracting one equation from the other to eliminate the variable.

The solution to the system of linear equations is a = 3 and c = 4. This solution can be verified by plugging the values of a and c back into the original equations to ensure that they are true.

Final Answer

The final answer is B. (3, -4).

Introduction

Systems of linear equations are a fundamental concept in mathematics, particularly in algebra and geometry. They consist of two or more linear equations that are solved simultaneously to find the values of the variables involved. In this article, we will answer some frequently asked questions (FAQs) about 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 are solved simultaneously to find the values of the variables involved.

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

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

  • Substitution method
  • Elimination method
  • Graphical method
  • Matrices 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 to solve for the other variable.

Q: What is the elimination method?

A: The elimination method involves multiplying the two equations by necessary multiples such that the coefficients of either a or c are the same in both equations, and then subtracting one equation from the other to eliminate the variable.

Q: What is the graphical method?

A: The graphical method involves graphing the two equations on a coordinate plane and finding the point of intersection, which represents the solution to the system.

Q: What is the matrices method?

A: The matrices method involves representing the system of linear equations as a matrix and then using row operations to solve for the variables.

Q: How do I know which method to use?

A: The choice of method depends on the specific system of linear equations and the variables involved. Some methods may be more efficient or easier to use than others.

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 the solution in both equations
  • Not using the correct method for the specific system
  • Not simplifying the equations before solving
  • Not checking for extraneous solutions

Q: How do I check if the solution is correct?

A: To check if the solution is correct, plug the values of the variables back into both equations and check if they are true.

Q: What are some real-world applications of systems of linear equations?

A: Systems of linear equations have many real-world applications, including:

  • Physics and engineering: to solve problems involving motion, forces, and energies
  • Economics: to model economic systems and make predictions about future trends
  • Computer science: to solve problems involving algorithms and data structures
  • Biology: to model population growth and disease spread

Conclusion

In conclusion, systems of linear equations are a fundamental concept in mathematics, and there are several methods of solving them. By understanding the different methods and common mistakes to avoid, you can become proficient in solving systems of linear equations and apply them to real-world problems.

Final Answer

The final answer is B. (3, -4).