find the missing opperation

[jjessicca]

Please Help:

If a*b = a - 3b and a#b = 2a+3b then what are * and # ?

Thanks,
Jessica

 

[pka]

Please expand upon your question?
The operators # & * are what they are definined as.
So what is your question about them?

 

[jjessicca]

Question Clarified

OK-
This is the complete question from the book (Saxon Advanced Mathmatics Lesson 8 #29)

a*b = a - 3b and a#b = 2a+3b Evaluate the expression (4*3) # 5.

I don't know how to explain this problem to my students.
(not even sure what it is asking)

 

[pka]

a*b = a - 3b and a#b = 2a+3b Evaluate the expression (4*3) # 5.
You are given the definitions. Lets use them.
(4*3)=(4)−3(3), that is a=4 and b=3
So (4*3)=−5.

(4*3)#5=(−5)#5=2(−5)+3(5).

 

[Denis]

Please Help:
If a*b = a - 3b and a#b = 2a+3b then what are * and # ?
Thanks,
Jessica

Looks to me like you need to come up with a proper operand to replace * and #;
just a little examination indicates the division sign is required:

a / b = a - 3b
a = ab - 3b^2
ab - a = 3b^2
a = 3b^2 / (b - 1) [1]

a / b = 2a + 3b
a = 2ab + 3b^2
a - 2ab = 3b^2
a = 3b^2 / (1 - b) [2]

[1][2]: b - 1 = 1 - b : 2b = 2 : b = 1 : subs to get a

 

[soroban]

Re: Question Clarified

Hello, jjessicca!

This is an exercise intended to exercise your Imagination
. . and your ability to follow directions.

\(\displaystyle a\,*\,b\:=\:a\,-\,3b\)

\(\displaystyle a\,\#\,b\:=\:2a\,+\,3b\)

Evaluate: \(\displaystyle \;(4\,*\,3)\,\#\,5\)

The operation "*" means: first number minus 3 times the second.

The operation "#" means: twice the first number plus 3 times the second.


So \(\displaystyle 4\,*\,3\) means: \(\displaystyle 4\,-\,3(3)\:=\:-5\)


Then we have: \(\displaystyle \;(-5)\,\#\,5\), which means: \(\displaystyle \;2(-5)\,+\,3(5)\:=\:5\)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

You see, we can invent our own operations . . .

Examples:

. . \(\displaystyle \;a\,\otimes\,b \;=\;a^b\;\) (First number raised to the power of the second number)

. . \(\displaystyle a\,\nabla\,b\;=\;\log_b(a)\;\) (Log of the first number to the base of the second number)

. . \(\displaystyle a\,\bullet\,b\;=\;\frac{a}{b}\;\) (Divide the first by the second)
. . . . . . . . . . . . . Hey! .We already have a symbol for that: \(\displaystyle \div\;\;\) LOL!

 

[konfusus]

Please Help:

If a*b = a - 3b and a#b = 2a+3b then what are * and # ?

Thanks,
Jessica

hey jess,
u should stick with pka's explanation...

a(operand)b =a - 3b

just defines a function with two variables. it's the same as saying f(a,b) = a - 3b

same with a(operand)b =2a + 3b

this means g(a,b) = 2a + 3b


so, (4*3) # 5 means:

g( f(4,3) , 5) = g ( (4-3*3) , 5 ) = 2* (4-3*3) + 3*5 = 5

 

[jjessicca]

Thanks everyone!
I think I was just looking at this one wrong-
Makes total sense now