Select The Correct Ordered Pair For The System Of Equations:${ \left{ \begin{array}{l} 3x - 4y = -14 \ 3x + 2y = -2 \end{array} \right. }$A. (2, 2) B. (-2, 2) C. (6, 2) D. (4, 9)

Mar 12, 2025 by ADMIN 183 views

**Solving a System of Linear Equations: A Step-by-Step Guide**

Introduction

Solving a system of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. 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 to the system of equations.

The System of Equations

The system of equations we will be solving is:

{ \left\{ \begin{array}{l} 3x - 4y = -14 \\ 3x + 2y = -2 \end{array} \right. \}

Method of Substitution

One way to solve this system of equations is by using the method of substitution. This method involves solving one of the equations for one variable and then substituting that expression into the other equation.

Let's start by solving the first equation for x:

$3x - 4y = -14$

We can add 4y to both sides of the equation to get:

$3x = -14 + 4y$

Now, we can divide both sides of the equation by 3 to get:

$x = \frac{-14 + 4y}{3}$

Substituting into the Second Equation

Now that we have an expression for x, we can substitute it into the second equation:

$3x + 2y = -2$

Substituting $x = \frac{-14 + 4y}{3}$ into the second equation, we get:

$3(\frac{-14 + 4y}{3}) + 2y = -2$

Simplifying the Equation

We can simplify the equation by multiplying the 3 out of the parentheses:

$-14 + 4y + 2y = -2$

Combining Like Terms

We can combine the like terms on the left-hand side of the equation:

$-14 + 6y = -2$

Adding 14 to Both Sides

We can add 14 to both sides of the equation to get:

$6y = 12$

Dividing Both Sides by 6

We can divide both sides of the equation by 6 to get:

$y = 2$

Finding the Value of x

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

$x = \frac{-14 + 4y}{3}$

Substituting $y = 2$ into the expression for x, we get:

$x = \frac{-14 + 4(2)}{3}$

Simplifying the Expression

We can simplify the expression by multiplying the 4 and the 2:

$x = \frac{-14 + 8}{3}$

Combining Like Terms

We can combine the like terms on the left-hand side of the expression:

$x = \frac{-6}{3}$

Dividing Both Sides by 3

We can divide both sides of the expression by 3 to get:

$x = -2$

The Solution to the System of Equations

Therefore, the solution to the system of equations is:

$(x, y) = (-2, 2)$

Conclusion

In this article, we solved a system of two linear equations with two variables using the method of substitution. We found that the solution to the system of equations is $(x, y) = (-2, 2)$ . This is just one of the many methods that can be used to solve a system of linear equations, and it is an important concept in mathematics and its applications.

Discussion

The solution to the system of equations is not among the options provided in the problem. However, we can see that the solution we found is a valid solution to the system of equations. This highlights the importance of checking the solution to a system of equations to ensure that it satisfies both equations.

Answer

The correct answer is not among the options provided in the problem. However, the solution we found is:

$(x, y) = (-2, 2)$

This is not among the options provided in the problem, but it is a valid solution to the system of equations.

Final Answer

The final answer is: $\boxed{(-2, 2)}$

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.

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

A: There are several methods for solving a system of linear equations, including the method of substitution, the method of elimination, and the method of graphing.

Q: What is the method of substitution?

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

Q: What is the method of elimination?

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

Q: What is the method of graphing?

A: The method of graphing involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: How do I know which method to use?

A: The choice of method depends on the type of equations and the variables involved. For example, if the equations are simple and have a clear point of intersection, the method of graphing may be the best choice. If the equations are more complex, the method of substitution or elimination may be more effective.

Q: What are some common mistakes to avoid when solving a system of linear equations?

A: Some common mistakes to avoid include:

Not checking the solution to ensure that it satisfies both equations
Not following the correct order of operations
Not simplifying the equations before solving
Not using the correct method for the type of equations involved

Q: How do I check the solution to a system of linear equations?

A: To check the solution, substitute the values of the variables into both equations and ensure that they are true.

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

A: If you get a system of linear equations with no solution, it means that the equations are inconsistent and there is no value of the variables that can satisfy both equations.

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

A: If you get a system of linear equations with infinitely many solutions, it means that the equations are dependent and there are an infinite number of values of the variables that can satisfy both equations.

Q: How do I determine if a system of linear equations has a unique solution, no solution, or infinitely many solutions?

A: To determine the type of solution, you can use the following criteria:

If the equations are consistent and have a unique solution, the system has a unique solution.
If the equations are inconsistent, the system has no solution.
If the equations are dependent, the system has infinitely many solutions.

Q: What are some real-world applications of solving a system of linear equations?

A: Solving a system of linear equations has numerous real-world applications, including:

Physics: Solving systems of linear equations is used to model the motion of objects and to solve problems involving forces and velocities.
Engineering: Solving systems of linear equations is used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Solving systems of linear equations is used to model economic systems and to solve problems involving supply and demand.
Computer Science: Solving systems of linear equations is used in computer graphics and game development to create realistic simulations and animations.

Q: How can I practice solving systems of linear equations?

A: You can practice solving systems of linear equations by working through examples and exercises in a textbook or online resource. You can also try solving systems of linear equations on your own using a calculator or computer program.