Given `A= [(3,4),(4,-3)] and B= [(24),(7)]`, find the matrix X such that AX= B.

Answer

To solve the equation \( AX = B \) where \( A = \begin{pmatrix} 3 & 4 \\ 4 & -3 \end{pmatrix} \) and \( B = \begin{pmatrix} 24 \\ 7 \end{pmatrix} \), we will follow these steps:

### Step 1: Define Matrix X
Let \( X \) be a column matrix of the form:
\[ X = \begin{pmatrix} x \\ y \end{pmatrix} \]

### Step 2: Set Up the Equation
We need to multiply matrix \( A \) by matrix \( X \):
\[ AX = \begin{pmatrix} 3 & 4 \\ 4 & -3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \]
This results in:
\[ AX = \begin{pmatrix} 3x + 4y \\ 4x - 3y \end{pmatrix} \]

### Step 3: Equate to Matrix B
Set the result equal to matrix \( B \):
\[ \begin{pmatrix} 3x + 4y \\ 4x - 3y \end{pmatrix} = \begin{pmatrix} 24 \\ 7 \end{pmatrix} \]

### Step 4: Create the System of Equations
From the above equation, we can create two equations:
1. \( 3x + 4y = 24 \) (Equation 1)
2. \( 4x - 3y = 7 \) (Equation 2)

### Step 5: Solve the System of Equations
We can solve these equations simultaneously. Let's solve Equation 1 for \( y \):
\[ 4y = 24 - 3x \implies y = \frac{24 - 3x}{4} \]

Now substitute \( y \) in Equation 2:
\[ 4x - 3\left(\frac{24 - 3x}{4}\right) = 7 \]
Multiply through by 4 to eliminate the fraction:
\[ 16x - 3(24 - 3x) = 28 \]
Distributing the -3:
\[ 16x - 72 + 9x = 28 \]
Combine like terms:
\[ 25x - 72 = 28 \]
Add 72 to both sides:
\[ 25x = 100 \]
Divide by 25:
\[ x = 4 \]

### Step 6: Substitute Back to Find y
Now substitute \( x = 4 \) back into the equation for \( y \):
\[ y = \frac{24 - 3(4)}{4} = \frac{24 - 12}{4} = \frac{12}{4} = 3 \]

### Step 7: Write the Solution Matrix
Thus, the matrix \( X \) is:
\[ X = \begin{pmatrix} 4 \\ 3 \end{pmatrix} \]

### Final Answer
The matrix \( X \) such that \( AX = B \) is:
\[ X = \begin{pmatrix} 4 \\ 3 \end{pmatrix} \]