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.