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Bot GPT 3.5 1. To solve the system of equations −3x − 4y = −23 and 3x − 5y = 5, we can use the method of elimination.
First, we multiply the second equation by -1 to make the coefficients of x the same magnitude but with opposite signs:
-1(3x - 5y) = -1(5)
-3x + 5y = -5
Now we can add the two equations together:
(-3x - 4y) + (-3x + 5y) = (-23) + (-5)
-6x + y = -28
To eliminate the y variable, we can multiply the first equation by 5 and the second equation by 4:
5(-3x - 4y) = 5(-23)
-15x - 20y = -115
4(-6x + y) = 4(-28)
-24x + 4y = -112
Adding the two equations together:
(-15x - 20y) + (-24x + 4y) = (-115) + (-112)
-39x - 16y = -227
Now we have a new equation:
-39x - 16y = -227
To solve for x, we can divide the equation by -39:
-39x/-39 - 16y/-39 = -227/-39
x + (16/39)y = 227/39
So the solution for x in terms of y is:
x = (227/39) - (16/39)y
To find y, we can substitute this expression for x into any of the original equations. Let's use the first equation:
-3x - 4y = -23
-3[(227/39) - (16/39)y] - 4y = -23
Simplifying:
(-3)(227/39) + (3)(16/39)y - 4y = -23
(-681/39) + (48/39)y - 4y = -23
(-681 + 48y - 156y)/39 = -23
(-681 - 108y)/39 = -23
-681 - 108y = (-23)(39)
-681 - 108y = -897
Now we can solve for y:
-108y = -897 + 681
-108y = -216
y = (-216)/(-108)
y = 2
Now we can substitute y = 2 into the expression for x:
x = (227/39) - (16/39)(2)
x = (227/39) - (32/39)
x = (227 - 32)/39
x = 195/39
x = 5
So the solution to the system of equations −3x − 4y = −23 and 3x − 5y = 5 is x = 5 and y = 2.
2. The system of equations 8x − 6y = 48 and 8x + 6y = 48 has infinitely many solutions. This is because the two equations represent parallel lines with the same slope and different y-intercepts. The two lines will never intersect, so any value of x and y that satisfies one equation will also satisfy the other.
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