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To solve this problem, we need to express the green vector \( \mathbf{x} \) as a linear combination of the two blue vectors \( \mathbf{a} \) and \( \mathbf{b} \). The equation would take the form:

\[
\mathbf{x} = c_1 \mathbf{a} + c_2 \mathbf{b}
\]

Where \( c_1 \) and \( c_2 \) are scalars (coefficients) for the vectors \( \mathbf{a} \) and \( \mathbf{b} \) respectively.

### Step-by-Step Approach:

1. **Identify the vectors:**
   - Vector \( \mathbf{a} \) appears to point in the positive x-direction (right), and its coordinates are approximately \( \mathbf{a} = (3, 0) \).
   - Vector \( \mathbf{b} \) points downward and to the left, with coordinates \( \mathbf{b} = (-1, -2) \).

2. **Coordinates of \( \mathbf{x} \):**
   - The green vector \( \mathbf{x} \) appears to point upward to the left, approximately at \( \mathbf{x} = (-1, 4) \).

3. **Set up the linear combination:**
   We need to solve for \( c_1 \) and \( c_2 \) in the following system:
   \[
   c_1 \mathbf{a} + c_2 \mathbf{b} = \mathbf{x}
   \]
   Substituting the values for the vectors:
   \[
   c_1 (3, 0) + c_2 (-1, -2) = (-1, 4)
   \]

4. **Break it into components:**
   - For the x-component: \( 3c_1 - c_2 = -1 \)
   - For the y-component: \( -2c_2 = 4 \)

5. **Solve for \( c_2 \):**
   From the second equation:
   \[
   -2c_2 = 4 \quad \Rightarrow \quad c_2 = -2
   \]

6. **Solve for \( c_1 \):**
   Substitute \( c_2 = -2 \) into the first equation:
   \[
   3c_1 - (-2) = -1 \quad \Rightarrow \quad 3c_1 + 2 = -1
   \]
   \[
   3c_1 = -3 \quad \Rightarrow \quad c_1 = -1
   \]

### Final Answer:
The vector \( \mathbf{x} \) can be written as:
\[
\mathbf{x} = -1 \cdot \mathbf{a} + (-2) \cdot \mathbf{b}
\]
Or:
\[
\mathbf{x} = -\mathbf{a} - 2\mathbf{b}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you verify a linear combination of vectors?
2. What is the geometric meaning of a linear combination of two vectors?
3. How can we express a vector in different bases?
4. How would you represent three vectors in 3D space as a linear combination?
5. What are the conditions for vectors to form a linear basis?

**Tip:** When working with linear combinations, always ensure the vectors are linearly independent (i.e., they don’t lie on the same line) to guarantee that the system has a unique solution.

Write x as a linear combination of a and b.

This problem explains how to express a vector as a linear combination of two other vectors in 2D space. It involves solving a system of linear equations to find the coefficients that describe the vector x as a combination of vectors a and b. Suitable for high school students in Grades 10-12.

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