Final answer:
The system of equations -64x + 96y = 176 and 56x - 84y = -147 simplifies to two distinct linear equations with different slopes, indicating they will intersect at one point. Thus, the system has one solution.
Step-by-step explanation:
To determine how many solutions the system of equations has, we can either solve the system or analyze the coefficients of the variables. The system of equations is –-64x + 96y = 176 and 56x – 84y = -147. Both equations can be simplified by dividing by their greatest common divisor to find out if they are multiples of each other (or if they represent the same line).
For the first equation, the greatest common divisor of 64 and 96 is 32:
- –(64/32)x + (96/32)y = (176/32)
- –-2x + 3y = 5.5
For the second equation, the greatest common divisor of 56 and 84 is 28:
- (56/28)x – (84/28)y = (-147/28)
- 2x – 3y = -5.25
After simplifying, we can see that the two equations are not multiples of each other. Instead, they have different slopes, indicating that they will intersect at one point. Therefore, the system has one solution.