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An ordered pair(a, b) for which the system of linear equations (1+ a)x +by +z=2 ax +(1+ b)y +z=3 and ax +by +2z=2 has a unique solution is?
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An ordered pair(a, b) for which the system of linear equations (1+ a)x...
Solution:
To find the ordered pair (a, b) for which the given system of linear equations has a unique solution, we can use the determinant method.
Determinant Method:
The system of linear equations can be written in matrix form as:
$\begin{bmatrix}
1 & a & -b \\
a & 1-b & 1 \\
a & b & 2 \\
\end{bmatrix}$
$\begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix}$ =
$\begin{bmatrix}
2 \\
3 \\
2 \\
\end{bmatrix}$
The system has a unique solution if and only if the determinant of the coefficient matrix is non-zero.
So, we need to find the determinant of the matrix:
$\begin{vmatrix}
1 & a & -b \\
a & 1-b & 1 \\
a & b & 2 \\
\end{vmatrix}$
Expanding along the first row, we get:
$\begin{vmatrix}
1-b & 1 \\
b & 2 \\
\end{vmatrix}$ - a $\begin{vmatrix}
a & 1 \\
b & 2 \\
\end{vmatrix}$ + b $\begin{vmatrix}
a & 1-b \\
a & b \\
\end{vmatrix}$
Simplifying, we get:
$(1-b)(2) - b(1) - a(2b-1) + b(a-b)$
$= 2-2b - 2ab + b^2$
$= (b-1)^2 - 2ab$
For the system to have a unique solution, this determinant must be non-zero.
So, we have two cases:
1. (b-1)^2 - 2ab = 0
In this case, the determinant is zero and the system either has no solution or infinitely many solutions.
2. (b-1)^2 - 2ab ≠ 0
In this case, the determinant is non-zero and the system has a unique solution.
Therefore, the ordered pair (a, b) for which the system has a unique solution is given by all values of a and b such that (b-1)^2 - 2ab ≠ 0.
To find the ordered pair (a, b) for which the given system of linear equations has a unique solution, we can use the determinant method.
Determinant Method:
The system of linear equations can be written in matrix form as:
$\begin{bmatrix}
1 & a & -b \\
a & 1-b & 1 \\
a & b & 2 \\
\end{bmatrix}$
$\begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix}$ =
$\begin{bmatrix}
2 \\
3 \\
2 \\
\end{bmatrix}$
The system has a unique solution if and only if the determinant of the coefficient matrix is non-zero.
So, we need to find the determinant of the matrix:
$\begin{vmatrix}
1 & a & -b \\
a & 1-b & 1 \\
a & b & 2 \\
\end{vmatrix}$
Expanding along the first row, we get:
$\begin{vmatrix}
1-b & 1 \\
b & 2 \\
\end{vmatrix}$ - a $\begin{vmatrix}
a & 1 \\
b & 2 \\
\end{vmatrix}$ + b $\begin{vmatrix}
a & 1-b \\
a & b \\
\end{vmatrix}$
Simplifying, we get:
$(1-b)(2) - b(1) - a(2b-1) + b(a-b)$
$= 2-2b - 2ab + b^2$
$= (b-1)^2 - 2ab$
For the system to have a unique solution, this determinant must be non-zero.
So, we have two cases:
1. (b-1)^2 - 2ab = 0
In this case, the determinant is zero and the system either has no solution or infinitely many solutions.
2. (b-1)^2 - 2ab ≠ 0
In this case, the determinant is non-zero and the system has a unique solution.
Therefore, the ordered pair (a, b) for which the system has a unique solution is given by all values of a and b such that (b-1)^2 - 2ab ≠ 0.
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An ordered pair(a, b) for which the system of linear equations (1+ a)x +by +z=2 ax +(1+ b)y +z=3 and ax +by +2z=2 has a unique solution is? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about An ordered pair(a, b) for which the system of linear equations (1+ a)x +by +z=2 ax +(1+ b)y +z=3 and ax +by +2z=2 has a unique solution is? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An ordered pair(a, b) for which the system of linear equations (1+ a)x +by +z=2 ax +(1+ b)y +z=3 and ax +by +2z=2 has a unique solution is?.
An ordered pair(a, b) for which the system of linear equations (1+ a)x +by +z=2 ax +(1+ b)y +z=3 and ax +by +2z=2 has a unique solution is? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about An ordered pair(a, b) for which the system of linear equations (1+ a)x +by +z=2 ax +(1+ b)y +z=3 and ax +by +2z=2 has a unique solution is? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An ordered pair(a, b) for which the system of linear equations (1+ a)x +by +z=2 ax +(1+ b)y +z=3 and ax +by +2z=2 has a unique solution is?.
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