Use matrices and elementary row operations to solve the following system of equations:
[tex]
\begin{aligned}
-x+2 y+3 z & =11 \
2 x-3 y & =-6 \
3 x-3 y+3 z & =3
\end{aligned}
[/tex]
Answer (2)
Write the augmented matrix.
Perform row operations to get the matrix in row-echelon form.
Use back-substitution to find the values of x , y , and z .
The solution is ( 3 , 4 , 2 ) .
Explanation
Problem Analysis We are given a system of three linear equations with three unknowns, x , y , and z :
− x + 2 y + 3 z 2 x − 3 y 3 x − 3 y + 3 z = 11 = − 6 = 3
Our goal is to solve this system using matrices and elementary row operations.
Forming the Augmented Matrix First, we write the augmented matrix for the system:
− 1 2 3 2 − 3 − 3 3 0 3 11 − 6 3
We will now perform elementary row operations to transform the matrix into row-echelon form.
Row Operations We multiply the first row by -1 to get a leading 1:
R 1 → − R 1
1 2 3 − 2 − 3 − 3 − 3 0 3 − 11 − 6 3
Next, we eliminate the 2 in the second row and the 3 in the third row:
R 2 → R 2 − 2 R 1 R 3 → R 3 − 3 R 1
1 0 0 − 2 1 3 − 3 6 12 − 11 16 36
Now, we eliminate the 3 in the third row:
R 3 → R 3 − 3 R 2
1 0 0 − 2 1 0 − 3 6 − 6 − 11 16 − 12
Finally, we divide the third row by -6 to get a leading 1:
R 3 → R 3 / − 6
1 0 0 − 2 1 0 − 3 6 1 − 11 16 2
Back-Substitution Now we use back-substitution to solve for x , y , and z .
From the third row, we have z = 2 .
From the second row, we have y + 6 z = 16 , so y + 6 ( 2 ) = 16 , which gives y = 16 − 12 = 4 .
From the first row, we have x − 2 y − 3 z = − 11 , so x − 2 ( 4 ) − 3 ( 2 ) = − 11 , which gives x − 8 − 6 = − 11 , so x = − 11 + 14 = 3 .
Thus, the solution is x = 3 , y = 4 , and z = 2 .
Final Answer Therefore, the solution to the system of equations is:
x y z = 3 = 4 = 2
We can write this as an ordered triple: ( 3 , 4 , 2 ) .
Examples
Systems of equations are used in various fields, such as engineering, economics, and computer science. For example, in electrical engineering, systems of equations can be used to analyze circuits and determine the current and voltage at different points. In economics, they can be used to model supply and demand curves and find equilibrium prices. In computer graphics, they are used to perform transformations on objects in 3D space. Understanding how to solve systems of equations is a fundamental skill in these areas, enabling professionals to model and solve real-world problems effectively. For instance, consider a business trying to optimize its production. They might use a system of equations to model the costs and revenues associated with producing different products, and then solve the system to find the optimal production levels that maximize profit. This demonstrates the practical application and importance of solving systems of equations.
To solve the given system of equations, we first formed the augmented matrix and used row operations to achieve row-echelon form. Through back-substitution, we found the solution to be (x, y, z) = (3, 4, 2). This process illustrates how to systematically solve a system of linear equations using matrices.
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