What Is The Solution \[$(a, C)\$\] To This System Of Linear Equations?$\[ \begin{aligned} 2a - 3c &= -6 \\ a + 2c &= 11 \end{aligned} \\]A. \[$\left(-\frac{76}{7}, \frac{17}{7}\right)\$\]B. \[$(3, -4)\$\]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 linear algebra. They consist of multiple 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, and we will use the method of substitution or elimination to find the solution.

The System of Linear Equations

The given system of linear equations is:

2a−3c=−6a+2c=11\begin{aligned} 2a - 3c &= -6 \\ a + 2c &= 11 \end{aligned}

We are asked to find the solution (a,c)(a, c) to this system of linear equations.

Method of Elimination

One way to solve this system of linear equations is by using the method of elimination. This method involves adding or subtracting the equations to eliminate one of the variables. In this case, we can multiply the first equation by 2 and the second equation by 3 to make the coefficients of aa in both equations equal.

4a−6c=−123a+6c=33\begin{aligned} 4a - 6c &= -12 \\ 3a + 6c &= 33 \end{aligned}

Now, we can add both equations to eliminate the variable aa.

(4a−6c)+(3a+6c)=−12+337a=21\begin{aligned} (4a - 6c) + (3a + 6c) &= -12 + 33 \\ 7a &= 21 \end{aligned}

Solving for aa, we get:

a=217a=3\begin{aligned} a &= \frac{21}{7} \\ a &= 3 \end{aligned}

Substitution Method

Another way to solve this system of linear equations is by using the substitution method. This method involves solving one of the equations for one variable and then substituting that expression into the other equation. In this case, we can solve the second equation for aa.

a=11−2c\begin{aligned} a &= 11 - 2c \end{aligned}

Now, we can substitute this expression for aa into the first equation.

2(11−2c)−3c=−622−4c−3c=−6−7c=−28\begin{aligned} 2(11 - 2c) - 3c &= -6 \\ 22 - 4c - 3c &= -6 \\ -7c &= -28 \end{aligned}

Solving for cc, we get:

c=−28−7c=4\begin{aligned} c &= \frac{-28}{-7} \\ c &= 4 \end{aligned}

Solution to the System of Linear Equations

Now that we have found the values of aa and cc, we can write the solution to the system of linear equations as:

(a,c)=(3,4)\begin{aligned} (a, c) &= (3, 4) \end{aligned}

Therefore, the solution to the system of linear equations is (3,4)(3, 4).

Conclusion

In this article, we have solved a system of two linear equations with two variables using the method of elimination and the substitution method. We have found the solution to the system of linear equations as (3,4)(3, 4). This solution satisfies both equations in the system, and it is the only solution to the system.

Discussion

The solution to the system of linear equations is unique, and it can be found using either the method of elimination or the substitution method. The method of elimination involves adding or subtracting the equations to eliminate one of the variables, while the substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation.

Final Answer

The final answer is: (3,4)\boxed{(3, 4)}

Introduction

Systems of linear equations are a fundamental concept in mathematics, particularly in algebra and linear algebra. 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 multiple linear equations that are solved simultaneously to find the values of the variables involved.

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

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

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

A: A system of linear equations consists of multiple linear equations, while a system of nonlinear equations consists of multiple nonlinear equations. Nonlinear equations are equations that are not linear, meaning that they cannot be written in the form ax + by = c.

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

A: There are several methods to solve a system of linear equations, including the method of elimination, the substitution method, and the graphing method. The method of elimination involves adding or subtracting the equations to eliminate one of the variables, while the substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation.

Q: What is the solution to a system of linear equations?

A: The solution to a system of linear equations is the set of values that satisfy all the equations in the system. In other words, it is the set of values that make all the equations true.

Q: Can a system of linear equations have multiple solutions?

A: No, a system of linear equations can have only one solution. If a system has multiple solutions, it is not a system of linear equations.

Q: How do I determine if a system of linear equations is consistent or inconsistent?

A: A system of linear equations is consistent if it has a solution, and it is inconsistent if it does not have a solution. To determine if a system is consistent or inconsistent, you can use the method of elimination or the substitution method to solve the system.

Q: What is the importance of systems of linear equations in real-life applications?

A: Systems of linear equations have numerous real-life applications, including physics, engineering, economics, and computer science. They are used to model real-world problems, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.

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 to solve systems of linear equations, including the method of elimination and the substitution method.

Q: How do I graph a system of linear equations?

A: To graph a system of linear equations, you can use a graphing calculator or a computer program. You can also use a coordinate plane to graph the equations and find the solution.

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

A: A system of linear equations is a set of multiple linear equations that are solved simultaneously to find the values of the variables involved, while a matrix equation is a single equation that involves a matrix and a vector.

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

A: Yes, you can use a matrix to solve a system of linear equations. A matrix can be used to represent the coefficients of the equations, and the solution to the system can be found by performing operations on the matrix.

Q: What is the importance of matrix equations in real-life applications?

A: Matrix equations have numerous real-life applications, including physics, engineering, economics, and computer science. They are used to model real-world problems, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.

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

A: Yes, you can use a computer program to solve a system of linear equations. Many computer programs, including MATLAB and Python, have built-in functions to solve systems of linear equations.

Q: How do I choose the best method to solve a system of linear equations?

A: The best method to solve a system of linear equations depends on the specific problem and the tools available. You can use the method of elimination, the substitution method, or a computer program to solve the system.

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

A: A system of linear equations consists of multiple linear equations, while a system of quadratic equations consists of multiple quadratic equations. Quadratic equations are equations that are quadratic in form, meaning that they can be written in the form ax^2 + bx + c = 0.

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

A: Yes, you can use a calculator to solve a system of quadratic equations. Many calculators have built-in functions to solve quadratic equations, including the quadratic formula.

Q: How do I graph a system of quadratic equations?

A: To graph a system of quadratic equations, you can use a graphing calculator or a computer program. You can also use a coordinate plane to graph the equations and find the solution.

Q: What is the importance of systems of quadratic equations in real-life applications?

A: Systems of quadratic equations have numerous real-life applications, including physics, engineering, economics, and computer science. They are used to model real-world problems, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.

Q: Can I use a matrix to solve a system of quadratic equations?

A: Yes, you can use a matrix to solve a system of quadratic equations. A matrix can be used to represent the coefficients of the equations, and the solution to the system can be found by performing operations on the matrix.

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

A: A system of linear equations consists of multiple linear equations, while a system of polynomial equations consists of multiple polynomial equations. Polynomial equations are equations that are polynomial in form, meaning that they can be written in the form ax^n + bx^(n-1) + ... + c = 0.

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

A: Yes, you can use a calculator to solve a system of polynomial equations. Many calculators have built-in functions to solve polynomial equations, including the rational root theorem and synthetic division.

Q: How do I graph a system of polynomial equations?

A: To graph a system of polynomial equations, you can use a graphing calculator or a computer program. You can also use a coordinate plane to graph the equations and find the solution.

Q: What is the importance of systems of polynomial equations in real-life applications?

A: Systems of polynomial equations have numerous real-life applications, including physics, engineering, economics, and computer science. They are used to model real-world problems, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.

Q: Can I use a matrix to solve a system of polynomial equations?

A: Yes, you can use a matrix to solve a system of polynomial equations. A matrix can be used to represent the coefficients of the equations, and the solution to the system can be found by performing operations on the matrix.

Final Answer

The final answer is: (3,4)\boxed{(3, 4)}