To solve the system of equations, we can use the method of elimination or substitution.
Let's use the method of elimination:
First, we need to multiply the first equation by 2 to match the coefficients of y:
-6x + 2y = -32
2(-6x + 2y) = 2(-32)
-12x + 4y = -64
Now, we can add this modified equation to the second equation to eliminate the x variable:
-12x + 4y = -64
+ 4x + 3y = -9
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-8x + 7y = -73
Now we can solve for y by multiplying the second equation by 8 and the first equation by 7 to eliminate the x variable again:
8(4x + 3y) = 8(-9)
32x + 24y = -72
7(-12x + 4y) = 7(-64)
-84x + 28y = -448
Now we can add these two equations to eliminate the x variable:
32x + 24y = -72
+ (-84x + 28y) = -448
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-52x + 52y = -520
Next, we can multiply the initial second equation by 6 to match the x coefficients:
6(4x + 3y) = 6(-9)
24x + 18y = -54
And now we can add this equation to the second equation to eliminate the y variable:
-52x + 52y = -520
+ (24x + 18y) = -54
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-28x - 30 = -574
To solve for x, we can divide both sides of the equation by -28:
(-28x - 30)/(-28) = (-574)/(-28)
x + (15/14) = 41/2
x = 41/2 - 15/14
x = (287 - 15)/14
x = 272/14
x = 136/7
Now that we have the value of x, we can substitute it into one of the original equations to solve for y. Let's use the first equation:
-6x + 2y = -32
-6(136/7) + 2y = -32
-816/7 + 2y = -32
2y = -32 + 816/7
2y = (-32 * 7 + 816)/7
2y = (224 - 32 + 816)/7
2y = 1008/7
y = (1008/7)/2
y = (1008/7)/(14/7)
y = 1008/14
y = 72/1
y = 72
Therefore, the solution to the system of equations is x = 136/7 and y = 72.