CLASIFY THE SYSTEM
{ X-Y+Z=6
{ 7X-2Y-2Z=1
{ 5X+6Y+3Z=11
A)CONSISTENT. ONE SOLUTION
B)INCONSISTENT. INFINITELY MANY SOLUTIONS
C)INCONSISTENT. NO SOLUTION
D)CONSISTENT. INFINITELY MANY SOLUTION
To classify the system of equations, you need to determine if it is consistent (has at least one solution) or inconsistent (has no solution). Additionally, if the system has infinitely many solutions, you need to decide if they are consistent or inconsistent.
One way to classify the system is by using Gaussian elimination or matrix operations. Let's solve the system using matrix operations (specifically, the augmented matrix) to determine its classification:
Step 1: Write down the augmented matrix representing the system:
⌈ 1 -1 1 | 6 ⌉
⌊ 7 -2 -2 | 1 ⌋
⌊ 5 6 3 | 11 ⌋
Step 2: Perform row operations to reduce the matrix to row-echelon or reduced row-echelon form. I'll use Gaussian elimination:
Multiply Row 1 by -7 and add to Row 2:
⌈ 1 -1 1 | 6 ⌉
⌊ 0 5 -9 | -41⌋
⌊ 5 6 3 | 11 ⌋
Multiply Row 1 by -5 and add to Row 3:
⌈ 1 -1 1 | 6 ⌉
⌊ 0 5 -9 | -41⌋
⌊ 0 11 -2 | -19⌋
Multiply Row 2 by -2 and add to Row 3:
⌈ 1 -1 1 | 6 ⌉
⌊ 0 5 -9 | -41⌋
⌊ 0 0 16 | -79⌋
Divide Row 3 by 16:
⌈ 1 -1 1 | 6 ⌉
⌊ 0 5 -9 | -41⌋
⌊ 0 0 1 | -79/16⌋
Step 3: Apply back substitution to obtain the values of the variables:
From the third row, we get:
Z = -79/16
Substituting Z into the second row, we get:
5Y - 9Z = -41
5Y - 9(-79/16) = -41
5Y + 711/16 = -41
5Y = -696/16
Y = -348/80
Y = -87/20
Substituting Y and Z into the first row, we get:
X - Y + Z = 6
X - (-87/20) + (-79/16) = 6
X + 87/20 - 79/16 = 6
X = 715/80
X = 143/16
Step 4: Determine the solutions:
The system has a unique solution since we found specific values for X, Y, and Z. Therefore, the system is classified as "A) CONSISTENT. ONE SOLUTION".
In summary, the system of equations is consistent with one solution. X = 143/16, Y = -87/20, and Z = -79/16.