[PorscheGAL]

Originally Posted by Bill Verburg

This type of question has been commonly showing up on HS Introductory algebra tests for decades

the # and * are abstract operations which are defined in the statement of the problem, other symbols, &,%,? etc. are commonly used as well, * is not multiplication this is mistake one.

The object of the question is to determine if the student can follow the abstract rules as presented w/o having any prior experience w/ them and then evaluate the resulting expression using the common math operations.

a * b = a - 3b is the definition of the * binary operator in terms of the usual subtraction binary operator and implicit binary multiplication using the variable values of a & b

similarly a # b = 2a+3b is the definition of the # binary operator in terms of the usual addition and implicit multiplication binary operators using the variable values of a & b.

the substitution principle, use of parenthesis and the use of the unary - operator are also involved in the solution.

starting w/ a=4, b =5 because it is inside parenthesis
a * b = a - 3b
becomes
4 * 3
4 - 3(3) per the def of * and substituting
4 - 9 regular arithmetic
-5

and then using the def of #
a # b = 2a+3b
(4 * 3) # 5 note that the current value of a was derived in the previous step as -5
(-5) # 5 implicit multiplication by 1 implies 1(-5) = -5
-5 # 5 the current value of a is -5 and the current value of b is 5
2(-5) +3(5) per the def of #
-10 + 15
5

Bill,

Thank you. That is the great explanation. Were you a teacher, perhaps? There is no doubt that after 20 years there is some things I don't remember. For the most, Kyle self-teaches and I come in to help if he has issues. We are definitely getting to the point in which that becomes a little harder. Luckily, he will have a professor teaching him next year.