Question6. Solve (if possible) each of the following systems of three equations in three variables using the method of GaussJordan elimination. (c)
Studdy Solution
STEP 1
1. The system of equations is given in three variables: , , and .
2. We will use the Gauss-Jordan elimination method to solve the system.
3. The system may have a unique solution, infinitely many solutions, or no solution.
STEP 2
1. Write the augmented matrix for the system of equations.
2. Use row operations to transform the matrix into reduced row-echelon form (RREF).
3. Interpret the RREF to find the solution(s) to the system.
STEP 3
Write the augmented matrix for the system:
STEP 4
Use row operations to transform the matrix into reduced row-echelon form (RREF).
First, eliminate the first column entries below the pivot in the first row:
Subtract 2 times the first row from the second row:
Subtract 3 times the first row from the third row:
The matrix becomes:
STEP 5
Next, make the pivot in the third row a 1 by dividing the third row by 5:
The matrix becomes:
STEP 6
Eliminate the third column entry in the first row by adding the third row to the first row:
The matrix becomes:
STEP 7
Interpret the RREF to find the solution(s) to the system.
The matrix corresponds to the following system of equations:
\begin{align*}
x_1 + 2x_2 &= 1 \\
x_3 &= -2
\end{align*}
Since the second row is all zeros, it does not provide any new information. We have a free variable, .
Express in terms of :
The solution is:
\begin{align*}
x_1 &= 1 - 2t \\
x_2 &= t \\
x_3 &= -2
\end{align*}
where is a parameter representing any real number.
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