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The given problem requires solving two simultaneous vector equations of the form:

\[
m\begin{pmatrix}-3 \\ 5\end{pmatrix} + n\begin{pmatrix}5 \\ 4\end{pmatrix} = \begin{pmatrix}14 \\ 9\end{pmatrix}
\]

This leads to two separate scalar equations, one for each component of the vectors. Let's break this down:

1. **First equation (from the first component):**
   \[
   -3m + 5n = 14
   \]

2. **Second equation (from the second component):**
   \[
   5m + 4n = 9
   \]

We now have the system of equations:
\[
\begin{aligned}
-3m + 5n &= 14 \\
5m + 4n &= 9
\end{aligned}
\]

### Step 1: Solve the system of equations.
We can solve this using substitution or elimination. I'll proceed with elimination by multiplying the equations to eliminate one variable.

- Multiply the first equation by 5 and the second equation by 3:
\[
-15m + 25n = 70
\]
\[
15m + 12n = 27
\]

### Step 2: Add the two equations to eliminate \(m\):
\[
-15m + 25n + 15m + 12n = 70 + 27
\]
\[
37n = 97
\]
\[
n = \frac{97}{37}
\]

### Step 3: Substitute \(n\) back into one of the original equations (let's use \(-3m + 5n = 14\)):
\[
-3m + 5\left(\frac{97}{37}\right) = 14
\]
\[
-3m + \frac{485}{37} = 14
\]
Multiply through by 37 to clear the fraction:
\[
-111m + 485 = 518
\]
\[
-111m = 518 - 485
\]
\[
-111m = 33
\]
\[
m = \frac{33}{-111} = -\frac{1}{3}
\]

Thus, the solution is:
\[
m = -\frac{1}{3}, \quad n = \frac{97}{37}
\]

Would you like further details or explanations on this solution? Here are some related questions to explore:

1. How do you solve simultaneous linear equations using substitution?
2. What is the geometric interpretation of solving vector equations?
3. How does elimination work for solving systems of linear equations?
4. Can you solve systems of equations using matrix methods (Gaussian elimination)?
5. How would the solution change if the vectors in the equation were three-dimensional?

**Tip:** Always check your solution by substituting the values of \(m\) and \(n\) back into both original equations to verify correctness.

Write down two simultaneous equations and solve them to find the value of m and the value of n.

Learn how to solve simultaneous vector equations using the elimination method. This problem involves breaking down a vector equation into two scalar equations and solving them step-by-step. Suitable for students in Grades 9-12.

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