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The given problem features a determinant equation involving two 3x3 matrices. The task likely involves proving the equality or solving for \(x\). Let's break down the problem step by step:

### Left Side of the Equation:
The left-hand side is the determinant of a 3x3 matrix:
\[
\begin{vmatrix}
a_1 + b_1 x & a_1 - b_1 x & c_1 \\
a_2 + b_2 x & a_2 - b_2 x & c_2 \\
a_3 + b_3 x & a_3 - b_3 x & c_3 \\
\end{vmatrix}
\]
We need to compute this determinant.

### Right Side of the Equation:
The right-hand side is expressed as:
\[
-2x \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}
\]
This involves the scalar \( -2x \) multiplied by the determinant of a different 3x3 matrix.

### Plan:
1. Compute the determinant on the left-hand side (LHS).
2. Compute the determinant on the right-hand side (RHS) after multiplication by \( -2x \).
3. Compare both sides and see if the equality holds, or solve for \( x \).

Let me now compute the determinants and verify the equality step-by-step.The determinants and their relationship are as follows:

- The determinant of the left-hand side (LHS) is:
  \[
  -2a_1b_2c_3x + 2a_1b_3c_2x + 2a_2b_1c_3x - 2a_2b_3c_1x - 2a_3b_1c_2x + 2a_3b_2c_1x
  \]

- The determinant of the right-hand side (RHS) is:
  \[
  -2x \left( a_1b_2c_3 - a_1b_3c_2 - a_2b_1c_3 + a_2b_3c_1 + a_3b_1c_2 - a_3b_2c_1 \right)
  \]

After simplification, both sides are equal, meaning the equation holds true.

Would you like more details on the steps involved in solving this, or any further clarifications?

Here are 5 related questions:
1. How is the determinant of a 3x3 matrix computed in general?
2. What are the properties of determinants involving scalar multiplication?
3. How does simplifying a determinant help in solving matrix equations?
4. Can you generalize this problem for matrices larger than 3x3?
5. How would the problem change if the matrix had different entries?

**Tip:** When proving matrix equalities, breaking them down into their determinants or other matrix operations often simplifies the problem.

Solve or prove the determinant equation involving two 3x3 matrices, as shown in the image, where the left-hand side matrix has variable terms involving x.

Explore how to solve a determinant equation featuring two 3x3 matrices with variable terms involving x. This problem covers key concepts in linear algebra, including determinant computation, matrix operations, and scalar multiplication. Suitable for undergraduate students or those studying linear algebra.

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