|
Question 615967: We're trying to help our soon with his math homework but we're totally stuck one this problem:
A store has a sale on almonds, pecans and pistachios. One lb of almonds, one lb of pecans and one lb of pistachios cost $12. Two lbs of almonds and three lbs of pecans cost $16. Three lbs of pecans and two lbs pistachios cost $24. Find the price of each kind of nut.
Thank you in advance for any help.
Found 3 solutions by scott8148, Theo, ikleyn:
Answer by scott8148(6628)   (Show Source):
You can put this solution on YOUR website! x=almond, y=pecan, z=pistachio
x + y + z = 12 ___ A
2x + 3y = 16 ___ B
3y + 2z = 24 ___ C
eliminating x ___ 2A - B ___ -y + 2z = 8 ___ D
eliminating y ___ 3D + C ___ 8z = 48
solve for z, then substitute back to find y and x
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! your answer is:
a = 2
b = 4
c = 6
a = pounds of almonds
b = pounds of pecans
c = pounds of pistachios
you have 3 equations that need to be solved simultaneously.
they are:
a + b + c = 12 (first equation)
2a + 3b = 16 (second equation)
3b + 2c = 24 (third equation)
take the first and second equation and solve them together to eliminate b.
multiply the first equation by 3 to get:
3a + 3b + 3c = 36
subtract second equation of 2a + 3b = 16 from it.
you get:
3a - 2a = a
3b - 3b = 0
3c - 0 = 3c
36 - 16 = 20
you are left with:
a + 3c = 20 (fourth equation)
take the first and third equations and solve them together to eliminate b.
multiply the first equation by 3 to get:
3a + 3b + 3c = 36
subtract third equation of 3b + 2c = 24 from it.
you get:
3a - 0 = 3a
3b - 3b = 0
3c - 2c = c
36 - 24 = 12
you are left with:
3a + c = 12 (fifth equation)
solve the fourth and fifth equations simultaneously to eliminate a.
you start with:
a + 3c = 20 (fourth equation)
3a + c = 12 (fifth equation)
multiply the fourth equation by 3 to get:
3a + 9c = 60
subtract the fifth equation of 3a + c = 12 from it.
you get:
3a - 3a = 0
9c - c = 8c
60 - 12 = 48
you are left with:
8c = 48
divide both sides of this equation by 8 to get:
c = 6
take the third equation to get:
3b + 2c = 24
substitute 6 for c in that equation and solve for b.
you get:
3b + 2*6 = 24 which becomes:
3b + 12 = 24 which becomes:
3b = 12 which becomes:
b = 4
take the first equation to get:
a + b + c = 12
substitute 6 for c and 4 for b to get:
a + 4 + 6 = 12
solve for a to get:
a = 2
your solutions are:
a = 2
b = 4
c = 6
substitute in all 3 original equations to confirm these solutions are good.
your original 3 equations are:
a + b + c = 12 (first equation)
2a + 3b = 16 (second equation)
3b + 2c = 24 (third equation)
after substitution of 2 for a and 4 for b and 6 for c, these equations become:
2 + 4 + 6 = 12
4 + 12 = 16
12 + 12 = 24
all these equation are true after substituting the values of a,b,c in them so the solutions are good because they solve all 3 equations simultaneously.
Answer by ikleyn(53250)  (Show Source):
You can put this solution on YOUR website! .
We're trying to help our  son with his math homework but we're totally stuck one this problem:
A store has a sale on almonds, pecans and pistachios. One lb of almonds, one lb of pecans and one lb of pistachios
cost $12. Two lbs of almonds and three lbs of pecans cost $16. Three lbs of pecans and two lbs pistachios cost $24.
Find the price of each kind of nut.
Thank you in advance for any help.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The solution by @Theo is tedious and boring, and you're unlikely to learn anything useful from his post.
Meanwhile, the problem has a much simpler solution, which teaches you to think and to approach the problem
in a new way. This solution is effective and instructive, and can work in many other problems.
Let 'a' be the pounds of almonds,
'b' be the pounds of pecans, and
'c' be the pounds of pistachios.
You have 3 equations that need to be solved simultaneously.
a + b + c = 12, (1)
2a + 3b = 16, (2)
3b + 2c = 24. (3)
This system of equations has a hidden symmetry, which will help us to solve.
Add equations (2) and (3) (both sides separately). You will get
2a + 6b + 2c = 40. (4)
From equation (4), subtract equation (1), multiplied by 2.
The terms with 'a' and 'c' will cancel, and you will get
6b - 2b = 40 - 2*12,
or
4b = 16.
Hence, b = 16/4 = 4.
Now, substitute b=4 into equation (2) and find 'a'
2a + 3*4 = 16 ---> 2a = 16 - 12 = 4 ---> a = 4/2 = 2.
Next, substitute b=4 into equation (3) and find 'c'
3*4 + 2c = 24 ---> 2c = 24 - 12 = 12 ---> c = 12/2 = 6.
At this point, the problem is solved completely.
ANSWER. a = $2 per pound (almonds); b = $4 per pound (pecans), and c = $6 per pound (pistachios).
Solved.
The lesson to learn
If you have to solve a system of three equations with three unknowns,
look if there is a hidden symmetry in it that could help.
|
|
|
| |
