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.