Solve The System Of Equations:$\[ \begin{aligned} x + Y + Z &= -6 \\ 2x + 4y + 2z &= -8 \\ -x + 8y - 3z &= 32 \end{aligned} \\]

Introduction

Solving a system of linear equations with three variables can be a challenging task, especially when dealing with complex equations. However, with the right approach and techniques, it is possible to find the solution to such systems. In this article, we will discuss how to solve a system of linear equations with three variables using the method of substitution and elimination.

The System of Equations

The system of equations we will be solving is given by:

$$\begin{aligned} x + y + z &= -6 \\ 2x + 4y + 2z &= -8 \\ -x + 8y - 3z &= 32 \end{aligned}$$

Method of Substitution

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

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

$$x = -6 - y - z$$

Now, substitute this expression for x into the second equation:

$$2(-6 - y - z) + 4y + 2z = -8$$

Expand and simplify the equation:

$$-12 - 2y - 2z + 4y + 2z = -8$$

Combine like terms:

$$-2y + 2z = 4$$

Now, substitute this expression for x into the third equation:

$$-(-6 - y - z) + 8y - 3z = 32$$

Expand and simplify the equation:

$$6 + y + z + 8y - 3z = 32$$

Combine like terms:

$$9y - 2z = 26$$

Method of Elimination

Another way to solve this system of equations is by using the method of elimination. This method involves adding or subtracting equations to eliminate one or more variables.

Let's start by multiplying the first equation by 2 and the second equation by 1:

$$2x + 2y + 2z = -12$$

$$2x + 4y + 2z = -8$$

Now, subtract the first equation from the second equation:

$$(2x + 4y + 2z) - (2x + 2y + 2z) = -8 - (-12)$$

Simplify the equation:

$$2y = 4$$

Now, divide both sides by 2:

$$y = 2$$

Finding the Values of x and z

Now that we have found the value of y, we can substitute it into one of the original equations to find the values of x and z.

Let's substitute y = 2 into the first equation:

$$x + 2 + z = -6$$

Simplify the equation:

$$x + z = -8$$

Now, substitute y = 2 into the third equation:

$$-x + 8(2) - 3z = 32$$

Simplify the equation:

$$-x + 16 - 3z = 32$$

Combine like terms:

$$-x - 3z = 16$$

Solving for x and z

Now we have two equations with two variables:

$$x + z = -8$$

$$-x - 3z = 16$$

Let's add the two equations to eliminate x:

$$(x + z) + (-x - 3z) = -8 + 16$$

Simplify the equation:

$$-2z = 8$$

Now, divide both sides by -2:

$$z = -4$$

Finding the Value of x

Now that we have found the value of z, we can substitute it into one of the equations to find the value of x.

Let's substitute z = -4 into the equation x + z = -8:

$$x + (-4) = -8$$

Simplify the equation:

$$x = -4$$

Conclusion

In this article, we have discussed how to solve a system of linear equations with three variables using the method of substitution and elimination. We have used the system of equations:

$$\begin{aligned} x + y + z &= -6 \\ 2x + 4y + 2z &= -8 \\ -x + 8y - 3z &= 32 \end{aligned}$$

to illustrate the methods. We have found the values of x, y, and z to be x = -4, y = 2, and z = -4.

Final Answer

The final answer is x = -4, y = 2, and z = -4.

Q: What is a system of linear equations with three variables?

A: A system of linear equations with three variables is a set of three linear equations that involve three variables, such as x, y, and z. Each equation is a linear combination of the variables, and the system is said to be consistent if it has a solution.

Q: What are the methods for solving systems of linear equations with three variables?

A: There are two main methods for solving systems of linear equations with three variables: the method of substitution and the method of elimination. The method of substitution involves solving one equation for one variable and then substituting that expression into the other equations. The method of elimination involves adding or subtracting equations to eliminate one or more variables.

Q: How do I choose between the method of substitution and the method of elimination?

A: The choice between the method of substitution and the method of elimination depends on the specific system of equations and the variables involved. If the system has a simple structure, such as a triangular or diagonal matrix, the method of substitution may be more efficient. However, if the system has a more complex structure, such as a non-square matrix, the method of elimination may be more efficient.

Q: What are some common mistakes to avoid when solving systems of linear equations with three variables?

A: Some common mistakes to avoid when solving systems of linear equations with three variables include:

• Not checking for consistency before solving the system
• 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 for consistency in a system of linear equations with three variables?

A: To check for consistency in a system of linear equations with three variables, you can use the following steps:

• Write the system of equations in matrix form
• Calculate the determinant of the coefficient matrix
• If the determinant is zero, the system is inconsistent
• If the determinant is non-zero, the system is consistent

Q: What is the significance of the determinant in solving systems of linear equations with three variables?

A: The determinant is a scalar value that can be used to determine the consistency of a system of linear equations with three variables. If the determinant is zero, the system is inconsistent, and there is no solution. If the determinant is non-zero, the system is consistent, and there is a unique solution.

Q: Can I use technology to solve systems of linear equations with three variables?

A: Yes, you can use technology to solve systems of linear equations with three variables. Many graphing calculators and computer algebra systems have built-in functions for solving systems of linear equations with three variables. You can also use software packages such as MATLAB or Python to solve systems of linear equations with three variables.

Q: What are some real-world applications of solving systems of linear equations with three variables?

A: Solving systems of linear equations with three variables has many real-world applications, including:

• Physics and engineering: Solving systems of linear equations with three variables is used to model physical systems, such as the motion of objects under the influence of gravity and friction.
• Economics: Solving systems of linear equations with three variables is used to model economic systems, such as the supply and demand of goods and services.
• Computer science: Solving systems of linear equations with three variables is used in computer graphics and game development to create 3D models and animations.

Conclusion

Solving systems of linear equations with three variables is an important topic in mathematics and has many real-world applications. By understanding the methods for solving these systems, including the method of substitution and the method of elimination, you can solve a wide range of problems in physics, engineering, economics, and computer science.