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The equation presented is a system of linear equations. Let's break it down and solve step by step.

The matrix equation is given as:

\[
\begin{pmatrix} 2 & 1 \\ 6 & -2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -1 \\ 7 \end{pmatrix}
\]

This translates into two linear equations:

1. \( 2x + y = -1 \)
2. \( 6x - 2y = 7 \)

We need to solve this system of equations for \( x \) and \( y \), and then substitute the values into \( 5x + 2y \).

### Solving the system of equations:

From the first equation:
\[
y = -1 - 2x
\]

Substitute this expression for \( y \) into the second equation:
\[
6x - 2(-1 - 2x) = 7
\]

Simplifying:
\[
6x + 2 + 4x = 7
\]
\[
10x + 2 = 7
\]
\[
10x = 5
\]
\[
x = \frac{1}{2}
\]

Now substitute \( x = \frac{1}{2} \) into the expression for \( y \):
\[
y = -1 - 2\left(\frac{1}{2}\right) = -1 - 1 = -2
\]

Thus, the solution is \( x = \frac{1}{2} \) and \( y = -2 \).

### Now calculate \( 5x + 2y \):
\[
5x + 2y = 5\left(\frac{1}{2}\right) + 2(-2) = \frac{5}{2} - 4 = \frac{5}{2} - \frac{8}{2} = \frac{-3}{2}
\]

Therefore, \( 5x + 2y = \frac{-3}{2} \).

Would you like further details or have any questions?

### Related questions:
1. How do you solve a system of linear equations using substitution?
2. What are other methods to solve systems of equations, like elimination or matrix inversion?
3. How do you multiply matrices and vectors in a general sense?
4. What are the properties of determinants in relation to solving linear equations?
5. How can you solve a system of equations using Gaussian elimination?

### Tip:
When solving systems of equations, always check your solution by plugging the values back into the original equations to verify their correctness.

Given the matrix equation (2 1) (x) = (-1), (6 -2) (y) (7), find 5x + 2y.

This problem involves solving a system of linear equations using matrix multiplication. The solution uses substitution to find the values of x and y, then calculates 5x + 2y. This problem is ideal for students in Grades 10-12.

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