Final answer:
To determine the number of solutions, we need to analyze the slopes of the lines represented by the equation. If the slopes are equal and the y-intercepts are different, the system has no solution. If the slopes and the y-intercepts are equal, the system has infinitely many solutions. The given system of equations has one unique solution.
Step-by-step explanation:
The given system of equations is 5x - 9y = 16.
To determine the number of solutions, we need to analyze the slopes of the lines represented by the equation. If the slopes are equal and the y-intercepts are different, the system has no solution. If the slopes and the y-intercepts are equal, the system has infinitely many solutions. And if the slopes are different, and the y-intercepts are different, the system has one unique solution.
Let's rearrange the equation into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:
- 5x - 9y = 16
- -9y = -5x + 16
- y = (5/9)x - 16/9
From the equation, we can see that the slope is 5/9 and the y-intercept is -16/9.
Since the slopes of the lines represented by the equation are different, and the y-intercepts are also different, the system has one unique solution.