Answer:
The last system is that one with exactly one solution:
–2x + 7y = –5
2x + 7y = –9
Explanation:
We can easily check the number of solutions that each system has in this problem, since the equations contain most of them combinations of terms in y and in x that can be easily cancelled out via simple term by term addition.
Going through them system by system, and adding term by term the equations which are in standard form we get:
(1) –5x – 10y = 24
5x + 10y = 16
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0x + 0y = 40 which sates a mathematical absurd: "0 = 40", and therefore there are no x and y solution values that could verify such. This also means that the lines represented by the system do NOT intersect (they are parallel lines)
(2) 3x + 4y = –7
–3x – 4y = 7
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0x + 0y = 0 which gives that zero equals zero, which any pair (x, y) belonging to any of the two equations can verify, so an infinite number of solutions (the lines in the system overlap)
(3) –12x + 8y = 16
12x – 8y = 16
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0x + 0y = 32 which sates a mathematical absurd: "0 = 2", and therefore there are no x and y solution values that could verify such. This also means that the lines represented by the system do NOT intersect (they are parallel lines)
(4) –2x + 7y = –5
2x + 7y = –9
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0x + 14y = -14
This is the only system that shows solution. It allows us first to solve for "y" in the resultant equation: 14 y = -14 giving us the value y = -14/14 = -1
and then using it the unique value for x can be determined:
2x + 7y = –9
2x + 7 (-1) = -9
2x -7 = -9
2x = -9 + 7
2x = -2
x = -2/2 = -1
Therefore, the pair (-1, -1) is the unique solution to the system