Final answer:
To determine the number of solutions for two lines, we calculate the slopes. The lines given in the question have different slopes, indicating they will intersect at exactly one point. Hence, the system of equations has one solution.
Step-by-step explanation:
To determine if a pair of lines have no solution, one solution, or an infinite number of solutions, we need to find the slopes of the lines.
If the slopes are equal and the y-intercepts are different, the lines are parallel and there is no solution because they never intersect. If the slopes are equal and the y-intercepts are also equal, the lines are identical and there is an infinite number of solutions.
If the slopes are different, there is exactly one solution where the lines intersect.
Let's find the slope of the first line, which passes through the points (0,2) and (9,-1). Using the formula for slope (m) which is m = (y2 - y1) / (x2 - x1), we get m = (-1 - 2) / (9 - 0) = -3/9 = -1/3.
For the second line, which passes through (12, 7) and (-6, -5), the slope is m = (-5 - 7) / (-6 - 12) = -12 / -18 = 2/3.
Since the slopes of the two lines are not equal (-1/3 ≠ 2/3), the lines are not parallel and will intersect in exactly one point. Therefore, the system of equations has one solution.