https://file.aiquickdraw.com/mathbot/file/1727150445401y3eeol2y.jpg

We are given the following matrices in the problem:

\[
B = \begin{bmatrix} 1 & -4 \\ 3 & 5 \end{bmatrix}, \quad C = \begin{bmatrix} 2 & -5 \\ 1 & 7 \end{bmatrix}
\]
and it is asked to find a matrix \( X \), where \( X - C = 2B \).

### Step 1: Rearrange the equation
We start with the given equation:
\[
X - C = 2B
\]
To isolate \( X \), add \( C \) to both sides:
\[
X = 2B + C
\]

### Step 2: Compute \( 2B \)
Now, let's compute \( 2B \):
\[
2B = 2 \times \begin{bmatrix} 1 & -4 \\ 3 & 5 \end{bmatrix} = \begin{bmatrix} 2 & -8 \\ 6 & 10 \end{bmatrix}
\]

### Step 3: Add \( 2B \) and \( C \)
Next, add the matrices \( 2B \) and \( C \):
\[
X = \begin{bmatrix} 2 & -8 \\ 6 & 10 \end{bmatrix} + \begin{bmatrix} 2 & -5 \\ 1 & 7 \end{bmatrix}
\]
\[
X = \begin{bmatrix} 2+2 & -8+(-5) \\ 6+1 & 10+7 \end{bmatrix} = \begin{bmatrix} 4 & -13 \\ 7 & 17 \end{bmatrix}
\]

Thus, the matrix \( X \) is:
\[
X = \begin{bmatrix} 4 & -13 \\ 7 & 17 \end{bmatrix}
\]

Let me know if you need further details or explanations.

### Related Questions:
1. What are the properties of matrix addition and scalar multiplication used here?
2. How do we multiply a matrix by a scalar?
3. What is the determinant of matrix \( X \) and how is it calculated?
4. Can we find the inverse of matrix \( X \), and if so, how?
5. How do we verify that matrix \( X \) satisfies the equation \( X - C = 2B \)?

### Tip:
When adding or subtracting matrices, always ensure the dimensions are compatible (both matrices must have the same number of rows and columns).

Given matrices B = [[1, -4], [3, 5]] and C = [[2, -5], [1, 7]], find a 2x2 matrix X such that X - C = 2B.

Explore the solution to find matrix X given B = [[1, -4], [3, 5]] and C = [[2, -5], [1, 7]] such that X - C equals 2 times B. This problem involves key linear algebra concepts including matrix addition and scalar multiplication.

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