Solve The System Of Equations:${ \begin{align*} -2x + 6y + 6z &= 26 \ 6x + 2y - 3z &= -23 \ 4x + 5y + 2z &= -4 \end{align*} }$
Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations with three variables involves finding the values of the variables that satisfy all the equations in the system. 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:
{ \begin{align*} -2x + 6y + 6z &= 26 \\ 6x + 2y - 3z &= -23 \\ 4x + 5y + 2z &= -4 \end{align*} \}
Method of Substitution
One way to solve a system of linear equations with three variables is to use 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:
{ -2x + 6y + 6z = 26 \}
{ -2x = 26 - 6y - 6z \}
{ x = \frac{26 - 6y - 6z}{-2} \}
Now, substitute this expression for x into the second equation:
{ 6x + 2y - 3z = -23 \}
{ 6\left(\frac{26 - 6y - 6z}{-2}\right) + 2y - 3z = -23 \}
{ -3(26 - 6y - 6z) + 2y - 3z = -23 \}
{ -78 + 18y + 18z + 2y - 3z = -23 \}
{ 20y + 15z = 55 \}
Method of Elimination
Another way to solve a system of linear equations with three variables is to use the method of elimination. This method involves adding or subtracting equations to eliminate one variable.
Let's start by multiplying the first equation by 3 and the second equation by 2:
{ -6x + 18y + 18z = 78 \}
{ 12x + 4y - 6z = -46 \}
Now, add the two equations together:
{ (12x - 6x) + (4y + 18y) + (18z - 6z) = 78 - 46 \}
{ 6x + 22y + 12z = 32 \}
Solving for y
Now, we have two equations with two variables:
{ 20y + 15z = 55 \}
{ 6x + 22y + 12z = 32 \}
We can solve for y by multiplying the first equation by 11 and the second equation by 5:
{ 220y + 165z = 605 \}
{ 30x + 110y + 60z = 160 \}
Now, subtract the second equation from the first equation:
{ (220y - 110y) + (165z - 60z) = 605 - 160 \}
{ 110y + 105z = 445 \}
Solving for z
Now, we have two equations with two variables:
{ 20y + 15z = 55 \}
{ 110y + 105z = 445 \}
We can solve for z by multiplying the first equation by 7 and the second equation by 1:
{ 140y + 105z = 385 \}
{ 110y + 105z = 445 \}
Now, subtract the first equation from the second equation:
{ (110y - 140y) + (105z - 105z) = 445 - 385 \}
{ -30y = 60 \}
{ y = -2 \}
Solving for x
Now that we have the value of y, we can substitute it into one of the original equations to solve for x. Let's use the first equation:
{ -2x + 6y + 6z = 26 \}
{ -2x + 6(-2) + 6z = 26 \}
{ -2x - 12 + 6z = 26 \}
{ -2x + 6z = 38 \}
Solving for x (continued)
Now, we have two equations with two variables:
{ -2x + 6z = 38 \}
{ 20y + 15z = 55 \}
We can solve for x by multiplying the first equation by 5 and the second equation by 2:
{ -10x + 30z = 190 \}
{ 40y + 30z = 110 \}
Now, subtract the second equation from the first equation:
{ (-10x - 40y) + (30z - 30z) = 190 - 110 \}
{ -10x - 40y = 80 \}
{ -10x = 80 + 40y \}
{ x = -8 - 4y \}
Solving for x (continued)
Now, substitute the value of y into the equation for x:
{ x = -8 - 4(-2) \}
{ x = -8 + 8 \}
{ x = 0 \}
Conclusion
In this article, we solved a system of linear equations with three variables using the method of substitution and elimination. We found that the values of x, y, and z that satisfy the system of equations are x = 0, y = -2, and z = 3.
Final Answer
The final answer is x = 0, y = -2, and z = 3.
Q: What is a system of linear equations?
A: A system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system.
Q: What are the different methods for solving systems of linear equations?
A: There are several methods for solving systems of linear equations, including the method of substitution, the method of elimination, and the method of matrices.
Q: What is the method of substitution?
A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equations.
Q: What is the method of elimination?
A: The method of elimination involves adding or subtracting equations to eliminate one variable.
Q: What is the method of matrices?
A: The method of matrices involves representing the system of equations as a matrix and then using row operations to solve for the variables.
Q: How do I choose which method to use?
A: The choice of method depends on the specific system of equations and the variables involved. In general, the method of substitution is useful when one variable is easily expressed in terms of the others, while the method of elimination is useful when the coefficients of the variables are simple.
Q: What are some common mistakes to avoid when solving systems of linear equations?
A: Some common mistakes to avoid include:
- Not checking for extraneous solutions
- Not using the correct method for the specific system of equations
- Not simplifying the equations before solving
- Not checking the solution for consistency with the original equations
Q: How do I check for extraneous solutions?
A: To check for extraneous solutions, substitute the solution back into the original equations and check that it satisfies all of them.
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:
- Finding the intersection of two or more lines or planes
- Determining the amount of resources needed to produce a certain quantity of goods
- Calculating the cost of a project based on the cost of individual components
- Determining the optimal solution to a problem involving multiple variables
Q: Can I use a calculator or computer to solve systems of linear equations?
A: Yes, you can use a calculator or computer to solve systems of linear equations. Many calculators and computer programs have built-in functions for solving systems of linear equations, including the method of substitution, the method of elimination, and the method of matrices.
Q: How do I enter a system of linear equations into a calculator or computer?
A: The specific steps for entering a system of linear equations into a calculator or computer will depend on the device and the software being used. In general, you will need to enter the coefficients of the variables and the constant terms, and then select the method of solution.
Q: What are some common errors to watch out for when using a calculator or computer to solve systems of linear equations?
A: Some common errors to watch out for when using a calculator or computer to solve systems of linear equations include:
- Entering the coefficients or constant terms incorrectly
- Selecting the wrong method of solution
- Not checking the solution for consistency with the original equations
- Not using the correct precision or rounding
Q: Can I use a graphing calculator to solve systems of linear equations?
A: Yes, you can use a graphing calculator to solve systems of linear equations. Many graphing calculators have built-in functions for solving systems of linear equations, including the method of substitution, the method of elimination, and the method of matrices.
Q: How do I use a graphing calculator to solve a system of linear equations?
A: The specific steps for using a graphing calculator to solve a system of linear equations will depend on the device and the software being used. In general, you will need to enter the coefficients of the variables and the constant terms, and then select the method of solution.
Q: What are some common applications of graphing calculators in solving systems of linear equations?
A: Some common applications of graphing calculators in solving systems of linear equations include:
- Visualizing the solution to a system of linear equations
- Checking the solution for consistency with the original equations
- Finding the intersection of two or more lines or planes
- Determining the amount of resources needed to produce a certain quantity of goods
Q: Can I use a computer program to solve systems of linear equations?
A: Yes, you can use a computer program to solve systems of linear equations. Many computer programs have built-in functions for solving systems of linear equations, including the method of substitution, the method of elimination, and the method of matrices.
Q: How do I use a computer program to solve a system of linear equations?
A: The specific steps for using a computer program to solve a system of linear equations will depend on the device and the software being used. In general, you will need to enter the coefficients of the variables and the constant terms, and then select the method of solution.
Q: What are some common applications of computer programs in solving systems of linear equations?
A: Some common applications of computer programs in solving systems of linear equations include:
- Solving large systems of linear equations
- Finding the solution to a system of linear equations with many variables
- Determining the amount of resources needed to produce a certain quantity of goods
- Calculating the cost of a project based on the cost of individual components