SOLUTION: Hello members and tutors of algebra.com, <p> I'm a high school student, in year 8 and am 13 years old. This question has propped up in my problem solving examination and I can

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Question 131069: Hello members and tutors of algebra.com,

I'm a high school student, in year 8 and am 13 years old. This question has propped up in my problem solving examination and I can never figure them out, particularly where there are multiple variables concerned. I have had a number of these, and I can remember it is possible to solve using diophantine equations. My teacher showed me a way to solve it that doesn't necessarily work for all of these types of problems and I'd like to know the alternative method. Here's the question and an explanation of how the answer would be obtained, is greatly appreciated:

"A cabbage and a cauliflower cost $3 in total, a cauliflower and a pumpkin cost $4 in total and a pumpkin and a cabbage cost $5 in total. Find the total cost of one of each vegetable."

I'd like to know how you approach this problem and please teach me as I'd like to know. Thankyou very much,

Jasmine
Found 2 solutions by checkley71, stanbon:
Answer by checkley71(8403) (Show Source):
You can put this solution on YOUR website!
x+y=3 or x=3-y
y+z=4 or y=4-z
x+z=5 or x=5-z
setting the x equations equal we get:
3-y=5-z
-y=5-3-z
y=-2+z
now set the y equations equal.
4-z=-2+z
-z-z=-2-4
-2z=-6
z=-6/-2
z=3 answer for the pumpkin.
y=-2+4
y=2 answer for the cauliflower.
x=3-2
x=1 answer for the cabbage.

Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website!
"A cabbage and a cauliflower cost $3 in total,
a cauliflower and a pumpkin cost $4 in total
and a pumpkin and a cabbage cost $5 in total.
Find the total cost of one of each vegetable."
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This is one way to work the problem; I'm sure there are others.
Let the cauliflower cost be "a"
Let the pumpkin cost be "b"
Let the cabbage cost be "c"
--------------------------
Three equations
a + 0b + c = 3
a + b + 0c = 4
0a + b + c = 5
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Subtract the 1st equation from the 2nd to get:
a + 0b + c = 3
0 + b - c = 1
0 + b + c = 5
------------------------
Subtract the 2nd equation from the 3rd to get:
a + 0b + c = 3
0 + b - c = 1
0 + 0 + 2c = 4
=====================
Now:
1st: solve for c and get c = 2 (# of cabbages is 2)
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2nd: substitute into the 2nd equation to get:
b-2 = 1
b = 3 (# of pumpkins is 3)
--------
3rd: substitute into the 1st equation to get:
a + 0 + 2 = 3
a = 1 (# of cauliflower is 1)
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Good wishes on your studies.
Cheers,
Stan H.