The following system of equations hasx + 3y + 3z = 2x + 4y + 3z = 1x + 3y + 4z = 2a)Infinite Solutionb)Trivial Solutionc)Unique Solutiond)No SolutionCorrect answer is option 'C'. Can you explain this answer?

Question

The following system of equations hasx + 3y + 3z = 2x + 4y + 3z = 1x + 3y + 4z = 2&#8203;a)Infinite Solutionb)Trivial Solutionc)Unique Solutiond)No SolutionCorrect answer is option 'C'. Can you explain this answer?

Accepted Answer

Let A = {(1,3,3) (1,4,3) (1,3,4)}
|A| = 1(16-9) -3(4-3) +3(3-4)
|A| = 1(7) -3(1) +3(-1)
= 7 - 3 - 3
= 1
Therefore, A is not equal to zero, it has unique solution.

Answer

Answer

Z=0 , Y = -1, X= 5

Answer

The determinant of the given equations is not equal to 0 and there exists a unique solution

Answer

To determine the solution to the given system of equations, let's analyze the equations one by one.

Given:
Equation 1: 3x + 3y + 3z = 2
Equation 2: 4x + 4y + 3z = 1
Equation 3: x + 3y + 4z = 2

To find the solution, we can use the method of elimination or substitution. Let's use the elimination method in this case.

Step 1: Multiply equations to eliminate variables
To eliminate the variable 'x', we can multiply Equation 1 by 4 and Equation 2 by 3:
Equation 1: 12x + 12y + 12z = 8
Equation 2: 12x + 12y + 9z = 3

Now, subtract Equation 2 from Equation 1:
(12x + 12y + 12z) - (12x + 12y + 9z) = 8 - 3
3z = 5

Step 2: Solve for the remaining variables
Now, we have the equation 3z = 5. Solving for 'z' gives:
z = 5/3

Step 3: Substitute the value of z into the other equations
Substituting z = 5/3 in Equation 1:
3x + 3y + 3 * (5/3) = 2
3x + 3y + 5 = 2
3x + 3y = -3

Substituting z = 5/3 in Equation 3:
x + 3y + 4 * (5/3) = 2
x + 3y + 20/3 = 2
x + 3y = -14/3

Step 4: Solve the remaining equations
Now, we have two equations:
3x + 3y = -3
x + 3y = -14/3

To solve these equations, subtract the second equation from the first equation:
(3x + 3y) - (x + 3y) = -3 - (-14/3)
2x = -3 + 14/3

Simplifying, we get:
2x = -9/3 + 14/3
2x = 5/3
x = 5/6

Substituting x = 5/6 in the second equation:
(5/6) + 3y = -14/3
3y = -14/3 - 5/6
3y = -28/6 - 5/6
3y = -33/6
y = -11/6

Step 5: Final Solution
We have found the values of x, y, and z:
x = 5/6
y = -11/6
z = 5/3

Therefore, the given system of equations has a unique solution.

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