We are given a system of two equations: $\frac{1}{2}x - 2 = y$ $2y - 8x = 10$ We need to determine which of the following statements are true: 1. The point $(2, -1)$ is a solution to the system of equations.

1. Problem Description

We are given a system of two equations:

$\frac{1}{2}x - 2 = y$

$2y - 8x = 10$

We need to determine which of the following statements are true:

1. The point $(2, -1)$ is a solution to the system of equations.

2. The point $(2, -3)$ is a solution to the system of equations.

3. The point $(1, 9)$ is a solution to the system of equations.

4. There is no solution to the system of equations.

2. Solution Steps

First, let's rewrite the first equation as:

$y = \frac{1}{2}x - 2$

Now, let's substitute this into the second equation:

$2(\frac{1}{2}x - 2) - 8x = 10$

$x - 4 - 8x = 10$

$-7x = 14$

$x = -2$

Now, substitute $x = -2$ back into the first equation to find $y$:

$y = \frac{1}{2}(-2) - 2$

$y = -1 - 2$

$y = -3$

So, the solution to the system of equations is $(-2, -3)$.

Now, let's check the given statements:

1. The point $(2, -1)$ is a solution:

Substituting $x = 2$ and $y = -1$ into the first equation:

$-1 = \frac{1}{2}(2) - 2 = 1 - 2 = -1$. This is true.

Substituting $x = 2$ and $y = -1$ into the second equation:

$2(-1) - 8(2) = -2 - 16 = -18 \ne 10$. This is false.

So, $(2, -1)$ is not a solution.

2. The point $(2, -3)$ is a solution:

As we found the solution to the system of equations is $(-2, -3)$, therefore $(2, -3)$ is not a solution.

3. The point $(1, 9)$ is a solution:

Substituting $x = 1$ and $y = 9$ into the first equation:

$9 = \frac{1}{2}(1) - 2 = \frac{1}{2} - 2 = -\frac{3}{2}$. This is false.

So, $(1, 9)$ is not a solution.

4. There is no solution to the system of equations.

This is false, as we found the solution to be $(-2, -3)$.

Let's check if $(-2, -3)$ is a solution for both equations:

First equation: $y = \frac{1}{2}x - 2$

$-3 = \frac{1}{2}(-2) - 2 = -1 - 2 = -3$. This is true.

Second equation: $2y - 8x = 10$

$2(-3) - 8(-2) = -6 + 16 = 10$. This is true.

Therefore the solution is indeed $(-2, -3)$.

None of the given points (2, -1), (2, -3), (1, 9) are solutions to the system.

Since the question asks us to select all true statements, and none of the given points are solutions, the system has exactly one solution, so it's false to say there is no solution.

None of the provided options are correct.

Let's rewrite the first equation as $x - 2y = 4$.

The second equation is $-8x + 2y = 10$.

Adding these two equations gives $-7x = 14$, so $x = -2$.

Then $-2 - 2y = 4$, so $-2y = 6$, and $y = -3$.

So the only solution is $(-2, -3)$.

Since none of the first three points are the solution, and there is a solution, then none of the given statements are true. There might be a misunderstanding of the question or a mistake in the provided options. However, let's examine the claim of no solution in more detail.

From the equations, we have:

1) $x - 2y = 4$

2) $-8x + 2y = 10$

Adding the equations, we get:

$-7x = 14$, thus $x = -2$.

Substituting $x = -2$ into the first equation gives $-2 - 2y = 4$, so $-2y = 6$ and $y = -3$.

The solution is $(-2, -3)$.

3. Final Answer

None of the statements are true. Therefore, none of the options should be selected.