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The equations x + 2y + 2z = 1 and 2x + 4 y + 4z = 9 have
- a)infinite solution
- b)only one solution
- c)only two solutions
- d)no solution
Correct answer is option 'D'. Can you explain this answer?
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The equations x + 2y + 2z = 1 and 2x + 4 y + 4z = 9 havea)infinite sol...
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The equations x + 2y + 2z = 1 and 2x + 4 y + 4z = 9 havea)infinite sol...
Solution:
Given equations are:
x + 2y + 2z = 1 …(i)
2x + 4y + 4z = 9 …(ii)
To check whether they have a solution or not, let’s try to solve them.
(ii) can be simplified as follows:
2x + 4y + 4z = 9
=> x + 2y + 2z = 4.5 …(iii)
Now, (i) and (iii) can be written as a matrix equation:
\begin{bmatrix}1&2&2\\1&2&2\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}1\\4.5\end{bmatrix}
The coefficient matrix [A] = \begin{bmatrix}1&2&2\\1&2&2\end{bmatrix] has rank 1 as its two rows are proportional to each other. Also, the augmented matrix [A|B] = \begin{bmatrix}1&2&2&1\\1&2&2&4.5\end{bmatrix] has rank 2. Therefore, the system of equations has no solution.
Hence, the correct option is (d) no solution.
Given equations are:
x + 2y + 2z = 1 …(i)
2x + 4y + 4z = 9 …(ii)
To check whether they have a solution or not, let’s try to solve them.
(ii) can be simplified as follows:
2x + 4y + 4z = 9
=> x + 2y + 2z = 4.5 …(iii)
Now, (i) and (iii) can be written as a matrix equation:
\begin{bmatrix}1&2&2\\1&2&2\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}1\\4.5\end{bmatrix}
The coefficient matrix [A] = \begin{bmatrix}1&2&2\\1&2&2\end{bmatrix] has rank 1 as its two rows are proportional to each other. Also, the augmented matrix [A|B] = \begin{bmatrix}1&2&2&1\\1&2&2&4.5\end{bmatrix] has rank 2. Therefore, the system of equations has no solution.
Hence, the correct option is (d) no solution.
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