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What stands for iff in math?
If and only if (necessary and sufficient condition).
If y equals 2x plus 1 and x equals 4y-2 what is the value of xy?
We havey = 2x + 1 = 2(4y-2) + 1 = 8y - 3 iff 7y = 3 iff y = 3/7 iff x = 4y - 2 = 12/7 - 2 = -2/7(using "iff" for the double-arrow "is equivalent to" symbol which my PC won't copy).Thus xy = (-2/7)(3/7) = -6/7.
If the sum of 6 times Jack's age and 5 times Larry's age is 63 Jack is 1 year less than 3 times as old as Larry?
The question is "How old are Jack and Larry?" I assume. Let Jack's age be J years and Larry's age be L years. The first part of the question tells us that 6J + 5L = 63. The second part tells us that J = 3L - 1. Thus 6J + 5L = 63 iff 6(3L-1) + 5L = 63 iff 18L - 6 + 5L = 63 iff 23L = 69 iff L = 3 iff J = 3L - 1 = 3x3 - 1 = 8 (using "iff" for the "is equivalent to" symbol). So Jack is 8 years old and Larry is 3 years old.
The sum of a number and its square is 2?
One -- Well, since we are talking about a second-degree equation, we should at least check if there might be another root (it could be the same). The equation is: x + x^2 = 2 iff x^2 + x - 2 = 0 iff (x+1/2)^2 - 2 - 1/4 = 0 iff (x+1/2)^2 = (3/2)^2 iff x+1/2 = +- 3/2 iff x = -2 or x = 1 Hence, the other solution is -2. Note that -2 + (-2)^2 = -2 + 4 = 2 as required. -- The difference of a number's square and its double is -1? Answer: one (and only one). Check this.
Two lines lie in one plane if and only if the lines are parallel?
Two lines are coplanar iff they are parallel OR intersect.
