Algebra Word Problem, help please?!?
Question: A potter makes two kinds of painted pots, A and B. It takes 2 pounds of clay to make pot A, and 3 pounds of clay to make pot B. It takes 2 bottles of paint to paint pot A, and 1 bottle of paint to paint pot B. If the potter has 26 pounds of clay and 14 bottles of paint, how many of pot A and pot B should be made so there is no waste?
I am so confused how to do this...I have like 100% in this class, so I know the material, but am befuddled by this bloody word problem! Could someone please explain how to do it? Thanks!
Comments
You need to use simultaneous equations, one for clay and one for paint.
Lets let x be the number of pot A, and y be the number of pot B
Paint: 14 bottles, 2 bottles for A, 1 bottle for B
14=2x +y
Clay: 26 pounds, 2pounds for each A, 3 pounds for each B
26=2x+3y
Take one equation from the other
26=2x+3y
- 14=2x+y
12=0x+2y
6=y
So, there are 6 of pot B. Substitute y=6 back into either equation and solve for x to work out the number of pots for A.
14=2x+y
14=2x+6
8=2x
4=x
So there are 4 of pot A, 6 of pot B.
Let X be the amount of A Pots
Let Y be the amount of B Pots
So 2x+3y= 26 (26 being total amount of clay)
And 2x+ y = 14 (14 being total amount of paint)
These are actually 2 linear lines, and you're actually trying to find the point of intersection. You can either graph it or solve it algebraically, as I will show you.
Take 1 equation and isolate a variable. I will take the second equation because it is a lot simpler.
2x+ y=14
2x=14-y (carried over)
2x-14=-y (isolation y)
-2x+14=y (making y positive to make life easier by dividing each side by -1)
Then we substitute -2x+14 into the first equation
2x+3y= 26
2x+ 3(-2x + 14) = 26 (I substituted)
2x -6x +42 = 26 (Distributed the 3 into the bracket)
-4x = -16 (Collected Like terms, Isolated variable and simplified)
x = 4 (divided each side by -4 to get a co-efficient of 1)
This x value shows how many Pot A's there are.
We finally substitute 4 as x into the 2nd equation
2x+ y=14
2(4) + y = 14 (substitute)
8 + y = 14
y = 6 ( isolated variable)
This y value tells us how many B pots there are
Therefore, the potter can make 4 pot As and 6 pot Bs with 26 pounds of clay and 14 bottles of paint. This method is called substitution. (Now you do the check but I'm too lazy cuz I know im correct :P)
2 pounds of clay for each pot A and 3 pounds of clay for each pot B. We're using all of 26 pounds of clay so we obtain the equation:
2A + 3B = 26
Similarly, 2 bottles of paint for A and 1 bottle of paint for B. We're using all of 14 bottles, so
2A + B = 14
You have system of linear equation. Using elimination method,
2A + 3B = 26
2A + B = 14
---------------------
2B = 12
B = 6
Substituting 6 in place of B in the second equation,
2A + B = 14
2A + 6 = 14
2A = 8
A = 4.
So it should be 4 pot A and 6 pot B.
pot A= four pots; eight pounds of clay and eight bottles of paint
pot B= six pots; eighteen pounds of clay and six bottles of paint