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The solution set for the equations x − y = −1 and 3x + y = 9 is the single point (2, 3). A solution of a linear system is an assignment of values to the variables ,, …, such that each of the equations is satisfied. The set of all possible solutions is called the solution set. [5]
For example: "An even number is an integer which is divisible by 2." An extensional definition instead lists all objects where the term applies. For example: "An even number is any one of the following integers: 0, 2, 4, 6, 8..., -2, -4, -8..." In logic, the extension of a predicate is the set of all things for which the predicate is true. [49]
The idea becomes clearer by considering the general series 1 − 2x + 3x 2 − 4x 3 + 5x 4 − 6x 5 + &c. that arises while expanding the expression 1 ⁄ (1+x) 2, which this series is indeed equal to after we set x = 1.
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The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
An example of a more complicated (although small enough to be written here) solution is the unique real root of x 5 − 5x + 12 = 0. Let a = √ 2φ −1, b = √ 2φ, and c = 4 √ 5, where φ = 1+ √ 5 / 2 is the golden ratio. Then the only real solution x = −1.84208... is given by
2,240 pounds (1,016.0 kilograms; 1.0160 metric tons) exactly 12% more than the 2,000 pounds of the North American short ton, being 20 long hundredweight (112 lb) rather than 20 short hundredweight (100 lb) the weight of 35 cubic feet (0.991 m 3) [2] of salt water with a density of 64 pounds per cubic foot (1.03 g/cm 3) [1]
The long or imperial hundredweight of 8 stone or 112 pounds (50.80 kg) is defined in the British imperial system. [2] Under both conventions, there are 20 hundredweight in a ton, producing a "short ton" of 2,000 pounds (907.2 kg) and a "long ton" of 2,240 pounds (1,016 kg).