Let the number = x.
Five times the difference from one is 5(x-1).
Twice the number increased by 3 could be either 2(x+3) or 2x+3.
First case:
5(x-1) = 2(x+3)
5x - 5 = 2x + 6 (multiply it out)
(5x - 2x) - 5 = (2x - 2x) + 6 (subtract 2x from both sides)
3x - 5 = 6 (combine terms)
3x - 5 + 5 = 6 + 5 (add 5 to both sides)
3x = 11 (add it up)
x(3/3) = 11/3 (divide both sides by 3)
x = 11/3
The difference from 1 is 8/3; 5 times that is 40/3 = 13 1/3.
The same number increased by 3 is 20/3; twice that is 40/3 = 13 1/3, the same result.
Second case:
5(x-1) = 2x+3
5x - 5 = 2x + 3
3x - 5 = 3
3x = 8
x = 8/3 (the explanation of each step is the same as above)
Five times the number's difference from 1 is 5(8/3 - 3/3) = 25/3
Two times the number is 16/3, and that increased by 3 is 16/3+9/3 = 25/3, the same result.
The trouble with trying to express algebraic statements in English is that there's a lot of ambiguity built into the English language (or any natural language, for that matter).
Five times the difference from one is 5(x-1).
Twice the number increased by 3 could be either 2(x+3) or 2x+3.
First case:
5(x-1) = 2(x+3)
5x - 5 = 2x + 6 (multiply it out)
(5x - 2x) - 5 = (2x - 2x) + 6 (subtract 2x from both sides)
3x - 5 = 6 (combine terms)
3x - 5 + 5 = 6 + 5 (add 5 to both sides)
3x = 11 (add it up)
x(3/3) = 11/3 (divide both sides by 3)
x = 11/3
The difference from 1 is 8/3; 5 times that is 40/3 = 13 1/3.
The same number increased by 3 is 20/3; twice that is 40/3 = 13 1/3, the same result.
Second case:
5(x-1) = 2x+3
5x - 5 = 2x + 3
3x - 5 = 3
3x = 8
x = 8/3 (the explanation of each step is the same as above)
Five times the number's difference from 1 is 5(8/3 - 3/3) = 25/3
Two times the number is 16/3, and that increased by 3 is 16/3+9/3 = 25/3, the same result.
The trouble with trying to express algebraic statements in English is that there's a lot of ambiguity built into the English language (or any natural language, for that matter).
