Given matrix `A=[{:(,5),(,-3):}]` and matrix `B=[{:(,-1),(,7):}]` find matrix X such that : A+2X=B.

Answer

To solve the equation \( A + 2X = B \) for the given matrices \( A \) and \( B \), we will follow these steps:

### Step 1: Write down the matrices
Given:
\[ A = \begin{pmatrix} 5 & -3 \end{pmatrix} \]
\[ B = \begin{pmatrix} -1 & 7 \end{pmatrix} \]

### Step 2: Set up the equation
We need to find matrix \( X \) such that:
\[ A + 2X = B \]

### Step 3: Rearrange the equation
To isolate \( 2X \), we can rearrange the equation:
\[ 2X = B - A \]

### Step 4: Substitute the values of \( A \) and \( B \)
Now, substitute the values of \( A \) and \( B \):
\[ 2X = \begin{pmatrix} -1 & 7 \end{pmatrix} - \begin{pmatrix} 5 & -3 \end{pmatrix} \]

### Step 5: Perform the matrix subtraction
Subtract the corresponding elements of the matrices:
\[ 2X = \begin{pmatrix} -1 - 5 & 7 - (-3) \end{pmatrix} \]
\[ 2X = \begin{pmatrix} -6 & 10 \end{pmatrix} \]

### Step 6: Divide by 2 to find \( X \)
Now, divide each element of the matrix \( 2X \) by 2 to find \( X \):
\[ X = \frac{1}{2} \begin{pmatrix} -6 & 10 \end{pmatrix} \]
\[ X = \begin{pmatrix} -3 & 5 \end{pmatrix} \]

### Final Answer
Thus, the matrix \( X \) is:
\[ X = \begin{pmatrix} -3 & 5 \end{pmatrix} \]

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