Two lines, C and D, are represented by the following equations:

Line C: y = x + 5
Line D: y = −2x − 1

Which of the following options shows the solution to the system of equations and explains why?

(−2, 3), because the point does not lie on any axis
(−2, 3), because both lines pass through this point
(−2, 3), because one of the lines passes through this point
(−2, 3), because the point lies between the two axes

To find the solution to the system of equations represented by Line C and Line D, we need to find the point at which the two lines intersect.

First, let's set the two equations equal to each other and try to solve for x:

x + 5 = -2x - 1

To solve for x, we can simplify the equation:

3x + 6 = 0

Subtracting 6 from both sides, we have:

3x = -6

Dividing both sides by 3, we get:

x = -2

Now that we have the x-coordinate, we can substitute it into either equation to find the y-coordinate. Let's use Line C:

y = x + 5

Substituting x = -2, we have:

y = -2 + 5

Simplifying, we get:

y = 3

So the point of intersection of Line C and Line D is (-2, 3).

Now let's examine the answer options:

Option 1: (-2, 3), because the point does not lie on any axis: This is not the correct explanation. The fact that the point does not lie on any axis is irrelevant to the solution.

Option 2: (-2, 3), because both lines pass through this point: This is the correct explanation. We found that the point of intersection is (-2, 3), and this option correctly states that both lines pass through this point.

Option 3: (-2, 3), because one of the lines passes through this point: This is not the correct explanation. Both lines pass through the point of intersection, not just one of them.

Option 4: (-2, 3), because the point lies between the two axes: This is not the correct explanation. The position of the point between the axes is not relevant to the solution.

Therefore, the correct answer is Option 2: (-2, 3), because both lines pass through this point.