https://file.aiquickdraw.com/mathbot/file/1725022322454mbql7ty0.jpg

To determine the value of \(x\) for which the given vectors are **not linearly independent**, we must identify the condition under which the vectors become **linearly dependent**.

The vectors given in the image are:
\[
\mathbf{v_1} = \begin{bmatrix} x \\ 3 \end{bmatrix}, \quad \mathbf{v_2} = \begin{bmatrix} 12 \\ -18 \end{bmatrix}.
\]

Two vectors \(\mathbf{v_1}\) and \(\mathbf{v_2}\) are **linearly dependent** if there exists a scalar \(\lambda\) such that:
\[
\mathbf{v_1} = \lambda \mathbf{v_2}.
\]

This means:
\[
\begin{bmatrix} x \\ 3 \end{bmatrix} = \lambda \begin{bmatrix} 12 \\ -18 \end{bmatrix}.
\]

Expanding this equation, we get two separate equations:
\[
x = 12\lambda, \quad 3 = -18\lambda.
\]

From the second equation:
\[
\lambda = \frac{3}{-18} = -\frac{1}{6}.
\]

Substitute \(\lambda = -\frac{1}{6}\) into the first equation:
\[
x = 12 \left(-\frac{1}{6}\right) = -2.
\]

Thus, the vectors are **not linearly independent** (i.e., they are linearly dependent) when \(x = -2\).

**Answer: \(-2\).**

Would you like more details on this solution or have any questions?

### Related Questions:
1. How can we determine if three vectors are linearly independent or dependent?
2. What is the geometric interpretation of linearly dependent vectors?
3. How do we find the determinant of a matrix formed by vectors to check independence?
4. What does it mean for vectors to span a vector space?
5. How does the concept of linear independence extend to higher dimensions?

**Tip:** Remember, two vectors are linearly dependent if they lie on the same line through the origin.

Learn how to determine the linear dependence of vectors given a variable x in a detailed step-by-step solution. This problem covers concepts in linear algebra, specifically vector dependence and scalar multiplication. Suitable for advanced high school students.

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