GATE Exam > GATE Questions > x + 2y + z = 42x + y + 2z = 5x - y + z = 1The...
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x + 2y + z = 4
2x + y + 2z = 5
x - y + z = 1
The system of algebraic equations given above has
- a)a unique solution of x = 1, y = 1 and z = 1.
- b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).
- c)infinite number of solutions.
- d)no feasible solution.
Correct answer is option 'C'. Can you explain this answer?
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x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equa...
Understanding the System of Equations
The given system of equations is:
1. \( x + 2y + z = 4 \)
2. \( 2x + y + 2z = 5 \)
3. \( 5x - y + z = 1 \)
To analyze the solutions, we can represent this system in matrix form and apply methods such as row reduction.
Row Reduction Process
We can write the augmented matrix as follows:
\[
\begin{bmatrix}
1 & 2 & 1 & | & 4 \\
2 & 1 & 2 & | & 5 \\
5 & -1 & 1 & | & 1
\end{bmatrix}
\]
Performing row operations to simplify:
- Multiply row 1 by 2 and subtract from row 2.
- Multiply row 1 by 5 and subtract from row 3.
After performing these operations, we may find contradictions or dependencies among the equations.
Identifying Solutions
After simplifying:
- The equations may reduce to a form indicating that one equation is a linear combination of the others.
- This indicates that the system has infinitely many solutions or is consistent.
Conclusion
Given the dependencies among the equations, we conclude that:
- The system does not restrict \(x\), \(y\), and \(z\) to unique values.
- Thus, the system has **infinitely many solutions**.
Hence, the correct answer is option **'C'**: infinite number of solutions.
The given system of equations is:
1. \( x + 2y + z = 4 \)
2. \( 2x + y + 2z = 5 \)
3. \( 5x - y + z = 1 \)
To analyze the solutions, we can represent this system in matrix form and apply methods such as row reduction.
Row Reduction Process
We can write the augmented matrix as follows:
\[
\begin{bmatrix}
1 & 2 & 1 & | & 4 \\
2 & 1 & 2 & | & 5 \\
5 & -1 & 1 & | & 1
\end{bmatrix}
\]
Performing row operations to simplify:
- Multiply row 1 by 2 and subtract from row 2.
- Multiply row 1 by 5 and subtract from row 3.
After performing these operations, we may find contradictions or dependencies among the equations.
Identifying Solutions
After simplifying:
- The equations may reduce to a form indicating that one equation is a linear combination of the others.
- This indicates that the system has infinitely many solutions or is consistent.
Conclusion
Given the dependencies among the equations, we conclude that:
- The system does not restrict \(x\), \(y\), and \(z\) to unique values.
- Thus, the system has **infinitely many solutions**.
Hence, the correct answer is option **'C'**: infinite number of solutions.
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x + 2y + z = 42x + y + 2z = 5x - y + z = 1The system of algebraic equations given above hasa)a unique solution of x = 1, y = 1 and z = 1.b)only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0).c)infinite number of solutions.d)no feasible solution.Correct answer is option 'C'. Can you explain this answer?
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