Solve the system by using Gaussian elimination or Gauss-Jordan elimination.
[tex]
\begin{aligned}
-2 x-5 y+6 z & = -33 \\
x+3 y+5 z & = 27 \\
5 x+4 y-5 z & =50
\end{aligned}
[/tex]
The solution set is { $\square$, $\square$, $\square$ }
Answer (2)
Represent the system of equations as an augmented matrix.
Apply Gaussian elimination to transform the matrix into reduced row-echelon form.
Read the solution directly from the reduced row-echelon form.
The solution set is x = 7 , y = 5 , z = 1 .
Explanation
Problem Analysis We are given a system of three linear equations in three variables, x , y , and z :
− 2 x − 5 y + 6 z x + 3 y + 5 z 5 x + 4 y − 5 z = − 33 = 27 = 50
Our goal is to solve this system for x , y , and z using Gaussian elimination or Gauss-Jordan elimination.
Setting up the Augmented Matrix We can represent the system of equations as an augmented matrix:
− 2 1 5 − 5 3 4 6 5 − 5 − 33 27 50
We will use elementary row operations to transform this matrix into reduced row-echelon form.
Applying Gaussian Elimination After performing Gaussian elimination (or Gauss-Jordan elimination) on the augmented matrix, we obtain the following reduced row-echelon form:
1 0 0 0 1 0 0 0 1 7 5 1
This matrix corresponds to the solution x = 7 , y = 5 , and z = 1 .
Finding the Solution Therefore, the solution to the system of equations is x = 7 , y = 5 , and z = 1 .
Examples
Systems of equations are used in various fields, such as engineering, economics, and computer science. For example, in structural engineering, systems of equations can be used to analyze the forces and stresses in a structure. In economics, they can be used to model supply and demand in a market. In computer graphics, they are used to perform transformations and projections of objects in 3D space. Solving systems of equations allows engineers and scientists to make predictions and optimize designs.
Using Gaussian elimination, we transformed the system of equations into an augmented matrix and then performed row operations to find the values of x, y, and z. The solution was determined to be x = 7, y = 5, and z = 1. Thus, the answer set is {7, 5, 1}.
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