One more than 8 times a number is the same as 12 times the number decreased by 3. find the number.

Put into equation form.
1+(8*n) OR 8n+1
(12*n) - 3 OR? 12*(n-3) * it is important to clarify when the 3 comes into play. Are you subtracting 3 from the number, or are you subtracting three from the product of 12 and the number?

either way. the next step is to set both equations equal to each other.

1.  8n+1 = 12n-3 (subtract 3 from the product)
OR
2.  8n+1 = 12(n-3) (subtract 3 from the number)

working with either on, your next goal is to isolate the number (your variable, n) to one side.

1. 8n+1 = 12n-3
subtract 1 from both sides (addition property of equality)
rewrite. 8n = 12n-4
subtract 12n from both sides (subtraction property of equality)
*NOTE: coefficients (numbers with attached variables) must be divided from their variable in order to be separated.
rewrite. -4n = -4
divide both sides by -4 (because you want to get rid of the coefficient of n, you will divide both sides by it.
ANSWER: n=1

2. 8n+1 = 12(n-3)
let's look at the mess on the right side, first. In order to simplify '12(n-3)', simply DISTRIBUTE 12 by multiplying it into the parenthesis.
*NOTE: Parenthesis show order, BUT you can't subtract or add from a variable if you don't know its value. Therefore, it makes sense to distribute the 12 before attempting to isolate the variable.
rewrite. 8n+1 = (12*n) + (12*-3)
simplify. 8n+1 = 12n + (-36)
rewrite (simplify). 8n+1 = 12n-36
*NOTE: There are several ways to isolate the variable, but you are likely to take similar steps. In effort to show variation, this example will feature an altered route.
add 36 to both sides. (addition property of equality)
rewrite. 8n+37 = 12n
subtract 8n from both sides. (subtraction property of equality)
rewrite. 37 = 4n
*NOTE: Usually, variables are featured on the right side of equal signs, you may choose to change the side of each FULL expression in its entirety in order to create a proper equality.
rewrite. 4n = 37
as done before, divide both sides by the coefficient of the variable (4).
rewrite. n = 37/4 (if you choose, you may write the decimal/improper fraction form of the answer).