Question

Find the value of x, y, and z from the following equation:
 I.\(\begin{bmatrix} 4&3&\\x&5\end{bmatrix}=\begin{bmatrix}y&z\\1&5\end{bmatrix}\)

  II. \(\begin{bmatrix}x+y&2\\5+z&xy\end{bmatrix}=\begin{bmatrix}6&2\\5&8\end{bmatrix}\)

   III. \(\begin{bmatrix}x+y+z\\x+z\\y+z\end{bmatrix}=\begin{bmatrix}9\\5\\7\end{bmatrix}\)

Answer 1

(i)  \(\begin{bmatrix} 4&3&\\x&5\end{bmatrix}=\begin{bmatrix}y&z\\1&5\end{bmatrix}\)As the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding elements, we get :x = 1, y = 4, and z = 3

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(ii) \(\begin{bmatrix}x+y&2\\5+z&xy\end{bmatrix}=\begin{bmatrix}6&2\\5&8\end{bmatrix}\) As the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding elements, we get:
x + y = 6, xy = 8, 5 + z = 5
Now, 5 + z = 5 \(\Rightarrow\) z = 0
We know that:
(x − y)²= (x + y)²− 4xy
\(\Rightarrow\) (x − y)² = 36 − 32 = 4
\(\Rightarrow\) x − y = ±2
Now, when x − y = 2 and x + y = 6, we get x= 4 and y = 2
When x − y = − 2 and x + y = 6, we get x = 2 and y = 4
∴x = 4, y = 2, and z = 0 or x = 2, y = 4, and z = 0

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(iii) \(\begin{bmatrix}x+y+z\\x+z\\y+z\end{bmatrix}=\begin{bmatrix}9\\5\\7\end{bmatrix}\) As the two matrices are equal, their corresponding elements are also equal.
Comparing the corresponding elements, we get:
x + y + z = 9 … (1)
x + z = 5 … (2)
y + z = 7 … (3)
From (1) and (2), we have:
y + 5 = 9
\(\Rightarrow\) y = 4
Then, from (3), we have:
4 + z = 7
\(\Rightarrow\) z = 3
∴ x + z = 5
\(\Rightarrow\) x = 2
∴ x = 2, y = 4, and z = 3