Find `x` and `y` respectively such that `[{:(x-y,3),(2x-y,2x+1):}]=[{:(5,3),(12,15):}]`

Options

A. a) 7,2
B. b) 2,7
C. c) 3,4
D. d) 2,9

Answer

To solve the equation given in the question, we need to find the values of `x` and `y` such that:

\[ \begin{pmatrix} x - y & 3 \\ 2x - y & 2x + 1 \end{pmatrix} = \begin{pmatrix} 5 & 3 \\ 12 & 15 \end{pmatrix} \]

### Step 1: Set up the equations from the matrix equality

Since the two matrices are equal, we can equate their corresponding elements:

1. \( x - y = 5 \) (Equation 1)
2. \( 3 = 3 \) (This is always true and does not provide any new information)
3. \( 2x - y = 12 \) (Equation 2)
4. \( 2x + 1 = 15 \) (Equation 3)

### Step 2: Solve Equation 3 for `x`

From Equation 3:

\[ 2x + 1 = 15 \]

Subtract 1 from both sides:

\[ 2x = 15 - 1 \]
\[ 2x = 14 \]

Now, divide both sides by 2:

\[ x = \frac{14}{2} = 7 \]

### Step 3: Substitute `x` into Equation 1 to find `y`

Now that we have found \( x = 7 \), we can substitute this value into Equation 1:

\[ x - y = 5 \]
\[ 7 - y = 5 \]

Now, solve for `y`:

\[ -y = 5 - 7 \]
\[ -y = -2 \]

Multiply both sides by -1:

\[ y = 2 \]

### Step 4: Conclusion

Thus, the values of \( x \) and \( y \) are:

\[ x = 7, \quad y = 2 \]

### Final Answer

Therefore, the correct option is \( (7, 2) \).

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