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The system of equation, 5x + 2y + z = 3,7x + 10y + 2z = 7,2x + 8y + z = 4 has
- a)One Solution
- b)Two Solutions
- c)No Solution
- d)Infinite Solutions
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The system of equation, 5x + 2y + z = 3,7x + 10y + 2z = 7,2x + 8y + z ...
The given system of equations are of the form AX = Bwhich represents the system of non homogeneous equations.
Augmented matrix is given by
by applying row transformation R2 → R2 − (R1+R3), we get
by applying row transformation R2 → R2 − (R1+R3), we get
so, rank of matrix = rank of augmented matrix = 2 < number of variables
∴ System has infinite solutions
Alternatively we can simply observe that second equation is redundant as it can be obtained from rest two. Hence we have 2 primary equations and 3 variables hence infinite solution.
Most Upvoted Answer
The system of equation, 5x + 2y + z = 3,7x + 10y + 2z = 7,2x + 8y + z ...
Understanding the System of Equations
The given system of equations is:
1. 5x + 2y + z = 3
2. 7x + 10y + 2z = 7
3. 2x + 8y + z = 4
To determine the nature of the solutions, we can analyze the equations in terms of their relationships.
Step 1: Convert to Matrix Form
We can rewrite the system in matrix form as:
| 5 2 1 | | x | | 3 |
| 7 10 2 | * | y | = | 7 |
| 2 8 1 | | z | | 4 |
This representation helps in identifying the coefficients and constants.
Step 2: Analyze the Coefficient Matrix
To find out if the system has one solution, no solutions, or infinite solutions, we can calculate the determinant of the coefficient matrix. If the determinant is zero, the system could have either no solutions or infinitely many solutions.
Step 3: Row Reduction
Applying row operations helps in simplifying the system. If we reach a point where one equation is a linear combination of others, it indicates that the equations are dependent.
Conclusion: Infinite Solutions
After performing the necessary steps, you may find that the equations are dependent, leading to an infinite number of solutions. This happens when at least one equation can be derived from the others.
Final Answer
Hence, the correct answer is option 'D': Infinite Solutions. This indicates that there are multiple sets of values for x, y, and z that satisfy all equations simultaneously.
The given system of equations is:
1. 5x + 2y + z = 3
2. 7x + 10y + 2z = 7
3. 2x + 8y + z = 4
To determine the nature of the solutions, we can analyze the equations in terms of their relationships.
Step 1: Convert to Matrix Form
We can rewrite the system in matrix form as:
| 5 2 1 | | x | | 3 |
| 7 10 2 | * | y | = | 7 |
| 2 8 1 | | z | | 4 |
This representation helps in identifying the coefficients and constants.
Step 2: Analyze the Coefficient Matrix
To find out if the system has one solution, no solutions, or infinite solutions, we can calculate the determinant of the coefficient matrix. If the determinant is zero, the system could have either no solutions or infinitely many solutions.
Step 3: Row Reduction
Applying row operations helps in simplifying the system. If we reach a point where one equation is a linear combination of others, it indicates that the equations are dependent.
Conclusion: Infinite Solutions
After performing the necessary steps, you may find that the equations are dependent, leading to an infinite number of solutions. This happens when at least one equation can be derived from the others.
Final Answer
Hence, the correct answer is option 'D': Infinite Solutions. This indicates that there are multiple sets of values for x, y, and z that satisfy all equations simultaneously.
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