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solve the given problem with step by step

The problem asks to define linear combination and to write the vector \(\mathbf{y}\) as a linear combination of the given vectors.

Given vectors:

\[
\mathbf{v}_1 = \begin{pmatrix} 2 \\ -3 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} 4 \\ 8 \end{pmatrix}, \quad \text{and} \quad \mathbf{y} = \begin{pmatrix} 7 \\ 5 \end{pmatrix}
\]

### Step 1: Understanding Linear Combination

A vector \(\mathbf{y}\) is said to be a linear combination of vectors \(\mathbf{v}_1\) and \(\mathbf{v}_2\) if there exist scalars \(c_1\) and \(c_2\) such that:

\[
\mathbf{y} = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2
\]

### Step 2: Set Up the Equation

We want to express \(\mathbf{y}\) as:

\[
\begin{pmatrix} 7 \\ 5 \end{pmatrix} = c_1 \begin{pmatrix} 2 \\ -3 \end{pmatrix} + c_2 \begin{pmatrix} 4 \\ 8 \end{pmatrix}
\]

### Step 3: Write the System of Equations

This leads to the system of linear equations:

\[
\begin{aligned}
2c_1 + 4c_2 &= 7 \quad \text{(1)} \\
-3c_1 + 8c_2 &= 5 \quad \text{(2)}
\end{aligned}
\]

### Step 4: Solve the System of Equations

Let's solve this system using substitution or elimination.

#### Elimination Method:
Multiply equation (1) by 3 and equation (2) by 2 to eliminate \(c_1\):

\[
\begin{aligned}
6c_1 + 12c_2 &= 21 \quad \text{(3)} \\
-6c_1 + 16c_2 &= 10 \quad \text{(4)}
\end{aligned}
\]

Add equations (3) and (4):

\[
28c_2 = 31 \implies c_2 = \frac{31}{28}
\]

Substitute \(c_2 = \frac{31}{28}\) back into equation (1):

\[
2c_1 + 4\left(\frac{31}{28}\right) = 7
\]

Simplifying:

\[
2c_1 + \frac{124}{28} = 7 \implies 2c_1 + \frac{31}{7} = 7
\]

Multiply through by 7 to clear the fraction:

\[
14c_1 + 31 = 49 \implies 14c_1 = 18 \implies c_1 = \frac{18}{14} = \frac{9}{7}
\]

### Final Answer:

The solution is:

\[
c_1 = \frac{9}{7}, \quad c_2 = \frac{31}{28}
\]

Therefore, the vector \(\mathbf{y}\) can be written as:

\[
\mathbf{y} = \frac{9}{7} \mathbf{v}_1 + \frac{31}{28} \mathbf{v}_2
\]

Would you like further details or have any questions?

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Here are some related questions to explore:
1. How do you determine if a set of vectors is linearly independent?
2. What is the geometric interpretation of a linear combination of two vectors?
3. How can the concept of linear combination be extended to three or more vectors?
4. What role does the matrix rank play in determining if a vector can be expressed as a linear combination of other vectors?
5. How does the concept of a linear span relate to linear combinations?

**Tip:** When solving systems of equations, always check your solution by substituting back into the original equations.

Understand how to define linear combination and solve vector equations step-by-step with a detailed solution. This problem covers linear algebra concepts, including linear combinations of vectors and solving systems of linear equations. Suitable for advanced high school students.

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