Apply Gauss Elimination method to solve
the following equation
2x + y + z = 10
3x +2y + 2 = 18
x+4y+9z=16
Answers
Answer:
Step-by-step explanation:
Carl Friedrich Gauss lived during the late 18th century and early 19th century, but he is still considered one of the most prolific mathematicians in history. His contributions to the science of mathematics and physics span fields such as algebra, number theory, analysis, differential geometry, astronomy, and optics, among others. His discoveries regarding matrix theory changed the way mathematicians have worked for the last two centuries.
We first encountered Gaussian elimination in Systems of Linear Equations: Two Variables. In this section, we will revisit this technique for solving systems, this time using matrices.
The Augmented Matrix of a System of Equations
A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix.
For example, consider the following \displaystyle 2\times 22×2 system of equations.
\displaystyle \begin{array}{l}3x+4y=7\\ 4x - 2y=5\end{array}
3x+4y=7
4x−2y=5
We can write this system as an augmented matrix:
[
3
4
4
−
2
|
7
5
]
We can also write a matrix containing just the coefficients. This is called the coefficient matrix.
\displaystyle \left[\begin{array}{cc}3& 4\\ 4& -2\end{array}\right][
3
4
4
−2
]
A three-by-three system of equations such as
3
x
−
y
−
z
=
0
x
+
y
=
5
2
x
−
3
z
=
2
has a coefficient matrix
⎡
⎢
⎣
3
−
1
−
1
1
1
0
2
0
−
3
⎤
⎥
⎦
and is represented by the augmented matrix
⎡
⎢
⎣
3
−
1
−
1
1
1
0
2
0
−
3
|
0
5
2
⎤
⎥
⎦
Notice that the matrix is written so that the variables line up in their own columns: x-terms go in the first column, y-terms in the second column, and z-terms in the third column. It is very important that each equation is written in standard form \displaystyle ax+by+cz=dax+by+cz=d so that the variables line up. When there is a missing variable term in an equation, the coefficient is 0