Bob has $X$ dollars now. If Pat gives Bob a dollar, Bob will have twice as many dollars as Pat. If Bob gave Pat a dollar instead, they both will have the same number of dollars. What is the value of $X?$
Let p = amount of money that Pat has.
Let b = amount of money that Bob has.
If Pat gives Bob a dollar, then Pat will have p - 1 dollars and Bob will have b + 1 dollars.
Since Bob will have twice as much money as Pat has: b + 1 = 2(p - 1).
If Bob gives Pat a dollar, then Pat will have p + 1 dollars and Bob will have b - 1 dollars.
Since they will then have the same amount of money: p + 1 = b - 1.
Putting these two equations together:
b + 1 = 2(p - 1) ---> b + 1 = 2p - 2 ---> 3 = 2p - b
b - 1 - p + 1 ---> -2 = p - b
2p - b = 3 ---> 2p - b = 3
p - b = -2 ---> x -1 ---> -p + b = 2
Adding down the columns: p = 5
Substituting back into an equation: b = 7
Let p = amount of money that Pat has.
Let b = amount of money that Bob has.
If Pat gives Bob a dollar, then Pat will have p - 1 dollars and Bob will have b + 1 dollars.
Since Bob will have twice as much money as Pat has: b + 1 = 2(p - 1).
If Bob gives Pat a dollar, then Pat will have p + 1 dollars and Bob will have b - 1 dollars.
Since they will then have the same amount of money: p + 1 = b - 1.
Putting these two equations together:
b + 1 = 2(p - 1) ---> b + 1 = 2p - 2 ---> 3 = 2p - b
b - 1 - p + 1 ---> -2 = p - b
2p - b = 3 ---> 2p - b = 3
p - b = -2 ---> x -1 ---> -p + b = 2
Adding down the columns: p = 5
Substituting back into an equation: b = 7