What can you determine about the solution(s) of the following system of equations? -64x + 96y = 176 56x - 84y = -147

Steps to Solve

1. Check if the equations are multiples of each other

To determine if the system has a unique solution, no solution, or infinitely many solutions, we can check the ratios of the coefficients of $x$ and $y$, and the constants.

2. Find the ratio of the coefficients of x

The ratio of the coefficients of $x$ is: $$ \frac{-64}{56} = -\frac{8}{7} $$

3. Find the ratio of the coefficients of y

The ratio of the coefficients of $y$ is: $$ \frac{96}{-84} = -\frac{24}{21} = -\frac{8}{7} $$

4. Find the ratio of the constants

The ratio of the constants is: $$ \frac{176}{-147} $$ Since $176 = 16 \times 11$ and $147 = 3 \times 49 = 3 \times 7^2$, there is no common factor to simplify the fraction.

5. Compare the Ratios

We see that the ratios of the coefficients of $x$ and $y$ are equal, i.e., $$ \frac{-64}{56} = \frac{96}{-84} = -\frac{8}{7} $$ However, the ratio of the constants is different. $$ \frac{176}{-147} \neq -\frac{8}{7} $$

6. Conclusion

Since the ratios of the coefficients of $x$ and $y$ are equal, but the ratio of the constants is different, the system has no solution. The two lines are parallel and distinct.