Solve the matrix equation 2XplusAequalsB for X if Aequalsleft bracket Start 3 By 2 Matrix 1st Row 1st Column negative 2 2nd Column negative 7 2nd Row 1st Column 0 2nd Column 7 3rd Row 1st Column negative 3 2nd Column 4 EndMatrix right bracket
  and Bequalsleft bracket Start 3 By 2 Matrix 1st Row 1st Column 6 2nd Column negative 5 2nd Row 1st Column 6 2nd Column 2 3rd Row 1st Column 1 2nd Column 6 EndMatrix right bracket
 .


To solve the matrix equation \( 2X + A = B \) for \( X \), we first need to isolate \( X \).

### Step-by-Step Solution:
1. **Matrix Equation:**
   \[
   2X + A = B
   \]
   
2. **Isolate \( X \):**
   Subtract \( A \) from both sides:
   \[
   2X = B - A
   \]
   
3. **Divide by 2:**
   \[
   X = \frac{1}{2}(B - A)
   \]

4. **Substitute the Given Matrices for \( A \) and \( B \):**

   \[
   A = \begin{bmatrix} -2 & -7 \\ 0 & 7 \\ -3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 6 & -5 \\ 6 & 2 \\ 1 & 6 \end{bmatrix}
   \]

5. **Calculate \( B - A \):**

   \[
   B - A = \begin{bmatrix} 6 - (-2) & -5 - (-7) \\ 6 - 0 & 2 - 7 \\ 1 - (-3) & 6 - 4 \end{bmatrix}
   = \begin{bmatrix} 8 & 2 \\ 6 & -5 \\ 4 & 2 \end{bmatrix}
   \]

6. **Calculate \( X \):**

   \[
   X = \frac{1}{2} \begin{bmatrix} 8 & 2 \\ 6 & -5 \\ 4 & 2 \end{bmatrix}
   = \begin{bmatrix} 4 & 1 \\ 3 & -2.5 \\ 2 & 1 \end{bmatrix}
   \]

### Final Answer:
\[
X = \begin{bmatrix} 4 & 1 \\ 3 & -2.5 \\ 2 & 1 \end{bmatrix}
\]

Would you like any more details on any step or further explanations? 

Here are some related questions you might find useful:
1. How do you perform matrix addition and subtraction?
2. How can scalar multiplication be applied to a matrix?
3. What properties of matrix operations are essential for solving matrix equations?
4. How would you solve a similar equation if \( A \) and \( B \) were square matrices?
5. What are some real-world applications of solving matrix equations?

**Tip:** Remember that in matrix operations, each element's position is crucial. The operations are carried out element-wise, which differs from traditional algebraic operations!

Solve the matrix equation 2X + A = B for X if A = [[-2, -7], [0, 7], [-3, 4]] and B = [[6, -5], [6, 2], [1, 6]].

Learn how to solve the matrix equation 2X + A = B for X, where A = [[-2, -7], [0, 7], [-3, 4]] and B = [[6, -5], [6, 2], [1, 6]]. This solution includes detailed steps, matrix subtraction, and scalar multiplication, suitable for students in Grades 10-12.

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