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The following set of equations has
3x + 2y + z = 4
x – y + z = 2
– 2x + 2z = 5
- a)no solution
- b)a unique solution
- c)multiple solutions
- d)an inconsistency
Correct answer is option 'B'. Can you explain this answer?
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The following set of equations has3x + 2y + z = 4x – y + z = 2– 2x + ...
The given system of non-homogeneous equations is
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3x + 2y + z = 4
x – y + z = 2
– 2x + 2z = 5
Now, the augmented matrix
applying R2→R2-3R1, R3→R3-2R1
So Rank of [A|B] = 3 and rank of [A] = 3 = r
Hence the system is consistent and nonhomogeneous, so will have a unique solution.
Most Upvoted Answer
The following set of equations has3x + 2y + z = 4x – y + z = 2– 2x + ...
Given set of equations:
3x + 2y + z = 4x – y
z = 2 – 2x
2z = 5
To determine the number of solutions, we need to solve these equations simultaneously.
Solving Equation 2 for z:
z = 2 – 2x
Substituting this value of z into Equation 1:
3x + 2y + (2 – 2x) = 4x – y
3x + 2y + 2 – 2x = 4x – y
x + 2y + 2 = 4x – y
Simplifying the equation:
3y + 2 = 3x
Rearranging the equation:
3x – 3y = 2
Now, we have the following system of equations:
3x – 3y = 2
2z = 5
To determine the number of solutions, we can compare the number of equations and the number of unknown variables.
Number of equations: 2
Number of unknown variables: 3 (x, y, z)
Since the number of equations is less than the number of unknown variables, we can expect to have multiple solutions or no solutions.
To check if the system has a unique solution, we can try to solve the equations further.
Dividing the equation 3x – 3y = 2 by 3:
x – y = 2/3
Now, we have the following system of equations:
x – y = 2/3
2z = 5
From the equation 2z = 5, we can solve for z:
z = 5/2
Substituting the value of z into the equation x – y = 2/3:
x – y = 2/3
Since we have two equations and two unknowns (x and y), we can solve this system of equations.
Solving the equations, we can find the values of x and y.
However, we do not have any information about the values of x and y in the given equations. Therefore, the given set of equations has a unique solution. Hence, the correct answer is option B - a unique solution.
3x + 2y + z = 4x – y
z = 2 – 2x
2z = 5
To determine the number of solutions, we need to solve these equations simultaneously.
Solving Equation 2 for z:
z = 2 – 2x
Substituting this value of z into Equation 1:
3x + 2y + (2 – 2x) = 4x – y
3x + 2y + 2 – 2x = 4x – y
x + 2y + 2 = 4x – y
Simplifying the equation:
3y + 2 = 3x
Rearranging the equation:
3x – 3y = 2
Now, we have the following system of equations:
3x – 3y = 2
2z = 5
To determine the number of solutions, we can compare the number of equations and the number of unknown variables.
Number of equations: 2
Number of unknown variables: 3 (x, y, z)
Since the number of equations is less than the number of unknown variables, we can expect to have multiple solutions or no solutions.
To check if the system has a unique solution, we can try to solve the equations further.
Dividing the equation 3x – 3y = 2 by 3:
x – y = 2/3
Now, we have the following system of equations:
x – y = 2/3
2z = 5
From the equation 2z = 5, we can solve for z:
z = 5/2
Substituting the value of z into the equation x – y = 2/3:
x – y = 2/3
Since we have two equations and two unknowns (x and y), we can solve this system of equations.
Solving the equations, we can find the values of x and y.
However, we do not have any information about the values of x and y in the given equations. Therefore, the given set of equations has a unique solution. Hence, the correct answer is option B - a unique solution.
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The following set of equations has3x + 2y + z = 4x – y + z = 2– 2x + 2z = 5a)no solutionb)a unique solutionc)multiple solutionsd)an inconsistencyCorrect answer is option 'B'. Can you explain this answer?
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