What Is The Solution To The Following System?$\[ \begin{cases} 4x + 3y - Z = -6 \\ 6x - Y + 3z = 12 \\ 8x + 2y + 4z = 6 \end{cases} \\]A. \[$x = 1, Y = -3, Z = -1\$\]B. \[$x = 1, Y = -3, Z = 1\$\]C. \[$x = 1, Y = 3, Z =
Introduction
In mathematics, 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. These equations are typically represented in the form of ax + by + cz = d, where a, b, c, and d are constants, and x, y, and z are the variables. In this article, we will focus on solving a system of three linear equations with three variables.
The System of Linear Equations
The system of linear equations we will be solving is given by:
{ \begin{cases} 4x + 3y - z = -6 \\ 6x - y + 3z = 12 \\ 8x + 2y + 4z = 6 \end{cases} \}
Method of Solution
To solve this system of linear equations, we can use the method of substitution or elimination. In this case, we will use the elimination method to find the values of x, y, and z.
Step 1: Multiply the Equations by Necessary Multiples
To eliminate one of the variables, we need to multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same.
Let's multiply the first equation by 1 and the second equation by 3.
{ \begin{cases} 4x + 3y - z = -6 \\ 18x - 3y + 9z = 36 \end{cases} \}
Step 2: Add the Two Equations
Now, we can add the two equations to eliminate the variable y.
{ (4x + 3y - z) + (18x - 3y + 9z) = (-6) + 36 \}
{ 22x + 8z = 30 \}
Step 3: Multiply the First Equation by 2 and the Third Equation by 3
To eliminate the variable y from the first and third equations, we need to multiply the first equation by 2 and the third equation by 3.
{ \begin{cases} 8x + 6y - 2z = -12 \\ 24x + 6y + 12z = 18 \end{cases} \}
Step 4: Add the Two Equations
Now, we can add the two equations to eliminate the variable y.
{ (8x + 6y - 2z) + (24x + 6y + 12z) = (-12) + 18 \}
{ 32x + 10z = 6 \}
Step 5: Solve the System of Two Equations
We now have a system of two equations with two variables.
{ \begin{cases} 22x + 8z = 30 \\ 32x + 10z = 6 \end{cases} \}
We can solve this system using the method of substitution or elimination. Let's use the elimination method.
Step 6: Multiply the First Equation by 5 and the Second Equation by 4
To eliminate the variable x, we need to multiply the first equation by 5 and the second equation by 4.
{ \begin{cases} 110x + 40z = 150 \\ 128x + 40z = 24 \end{cases} \}
Step 7: Subtract the Two Equations
Now, we can subtract the two equations to eliminate the variable z.
{ (110x + 40z) - (128x + 40z) = 150 - 24 \}
{ -18x = 126 \}
Step 8: Solve for x
Now, we can solve for x.
{ x = -\frac{126}{18} \}
{ x = -7 \}
Step 9: Substitute x into One of the Equations
Now that we have the value of x, we can substitute it into one of the equations to find the value of z.
Let's substitute x into the first equation.
{ 22(-7) + 8z = 30 \}
{ -154 + 8z = 30 \}
{ 8z = 184 \}
{ z = \frac{184}{8} \}
{ z = 23 \}
Step 10: Substitute x and z into One of the Equations
Now that we have the values of x and z, we can substitute them into one of the equations to find the value of y.
Let's substitute x and z into the first equation.
{ 4(-7) + 3y - 23 = -6 \}
{ -28 + 3y - 23 = -6 \}
{ -51 + 3y = -6 \}
{ 3y = 45 \}
{ y = \frac{45}{3} \}
{ y = 15 \}
Conclusion
In this article, we solved a system of three linear equations with three variables using the elimination method. We first multiplied the equations by necessary multiples to eliminate one of the variables, then added the two equations to eliminate the variable y. We repeated this process to eliminate the variable x and finally solved for the value of z. We then substituted the values of x and z into one of the equations to find the value of y. The solution to the system of linear equations is x = -7, y = 15, and z = 23.
Answer
The correct answer is:
{ \begin{cases} x = -7 \\ y = 15 \\ z = 23 \end{cases} \}
This answer is not among the options provided, but it is the correct solution to the system of linear equations.
Introduction
Solving systems of linear equations can be a challenging task, especially for those who are new to the concept. In this article, we will address some of the most 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 are solved simultaneously to find the values of the variables involved. These equations are typically represented in the form of ax + by + cz = d, where a, b, c, and d are constants, and x, y, and z are the variables.
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: This method involves substituting the value of one variable from one equation into another equation to solve for the other variables.
- Elimination method: This method involves adding or subtracting equations to eliminate one or more variables and solve for the remaining variables.
- Graphical method: This method involves graphing the equations on a coordinate plane and finding the point of intersection to solve for the variables.
- Matrix method: This method involves representing the system of equations as a matrix and using matrix operations to solve for the variables.
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 equations where each equation is linear, meaning that it can be written in the form of ax + by + cz = d. A system of nonlinear equations, on the other hand, is a set of equations where at least one equation is nonlinear, meaning that it cannot be written in the form of ax + by + cz = d.
Q: How do I determine which method to use when solving a system of linear equations?
A: The choice of method depends on the specific system of equations and the variables involved. If the system has a simple structure, such as two equations with two variables, the substitution or elimination method may be the most efficient. If the system has a more complex structure, such as three equations with three variables, the matrix method may be more suitable.
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 for consistency: Make sure that the system of equations is consistent, meaning that it has a solution.
- Not checking for uniqueness: Make sure that the solution is unique, meaning that there is only one solution.
- Not using the correct method: Choose the correct method for the specific system of equations.
- Not checking for errors: Double-check your work for errors, such as incorrect calculations or incorrect substitutions.
Q: How do I check if a system of linear equations has a solution?
A: To check if a system of linear equations has a solution, you can use the following methods:
- Check for consistency: Make sure that the system of equations is consistent, meaning that it has a solution.
- Check for uniqueness: Make sure that the solution is unique, meaning that there is only one solution.
- Use the rank-nullity theorem: The rank-nullity theorem states that the rank of the coefficient matrix plus the nullity of the coefficient matrix is equal to the number of variables. If the rank is equal to the number of variables, then the system has a unique solution.
Q: What are some real-world applications of solving systems of linear equations?
A: Solving systems of linear equations has many real-world applications, including:
- Physics and engineering: Solving systems of linear equations is used to model and solve problems in physics and engineering, such as motion, forces, and energy.
- Computer science: Solving systems of linear equations is used in computer science to solve problems in machine learning, data analysis, and optimization.
- Economics: Solving systems of linear equations is used in economics to model and solve problems in economics, such as supply and demand, and resource allocation.
Conclusion
Solving systems of linear equations is a fundamental concept in mathematics and has many real-world applications. By understanding the different methods of solving systems of linear equations and avoiding common mistakes, you can become proficient in solving these types of problems.