Question

If \[\begin{bmatrix} 2x - y & x + 2y \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 2 & 3 \end{bmatrix}\]
 then the value of $x$ and $y$ will be: 
 

• A. $x = 1, y = 1$
• B. $x = 2, y = 1$
• C. $x = \dfrac{1}{2}, y = \dfrac{1}{2}$
• D. $x = 1, y = \dfrac{1}{2}$

Hint:

When solving a system of linear equations from matrices, equate corresponding elements of the matrices to form a system of equations.

Answer 1

The Correct Option is A

We are given the matrix equation: 

\[\begin{bmatrix} 2x - y & x + 2y \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 2 & 3 \end{bmatrix}\]

Step 1: Compare corresponding elements of the matrices. 
From the equation, we can equate the corresponding elements of both matrices. 1. From the top-left element: \[ 2x - y = 1 \text{(Equation 1)}. \] 2. From the top-right element: \[ x + 2y = 3 \text{(Equation 2)}. \] 3. From the bottom-left element: \[ 2 = 2 \text{(This is true, so no further information here)}. \] 4. From the bottom-right element: \[ 3 = 3 \text{(This is true, so no further information here)}. \]

Step 2: Solve the system of equations. 
We now have the system of equations: 1. \(2x - y = 1\) 2. \(x + 2y = 3\) Let's solve this system. From Equation 1, solve for \(y\): \[ y = 2x - 1. \] Substitute this into Equation 2: \[ x + 2(2x - 1) = 3, \] \[ x + 4x - 2 = 3, \] \[ 5x = 5, \] \[ x = 1. \] Now substitute \(x = 1\) into the expression for \(y\): \[ y = 2(1) - 1 = 1. \]

Step 3: Conclusion. 
The solution to the system is \(x = 1\) and \(y = 1\), so the correct answer is (A) \(x = 1, y = 1\).