Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.minus, 6, x, plus, 5, y, equals, minus, 9−6x+5y=−9minus, 11, x, plus, 10, y, equals, minus, 17−11x+10y=−17
Question
Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.minus, 6, x, plus, 5, y, equals, minus, 9−6x+5y=−9minus, 11, x, plus, 10, y, equals, minus, 17−11x+10y=−17
Solution
To determine if the system of equations has no solutions, infinitely many solutions, or exactly one solution, we can use the method of comparison or substitution. However, in this case, it's easier to use the method of comparison.
The system of equations is:
-6x + 5y = -9 -11x + 10y = -17
First, we can simplify these equations by dividing the first equation by -1 and the second equation by -1 to make the x and y coefficients positive:
6x - 5y = 9 11x - 10y = 17
Next, we can multiply the first equation by 2 and the second equation by 1 to make the coefficients of y the same in both equations:
12x - 10y = 18 11x - 10y = 17
Now, we subtract the second equation from the first:
12x - 11x = 18 - 17 x = 1
Substitute x = 1 into the first equation:
6*1 - 5y = 9 6 - 5y = 9 -5y = 9 - 6 -5y = 3 y = 3 / -5 y = -0.6
So, the system of equations has exactly one solution: x = 1, y = -0.6.
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