3 simultaenous equations

[vinay1983]

Four friends Laura, Nancy, James and Simon buy some pencils, pens and erasers from a shop. Laura buys 5 pencils, 6 pens, and 1 eraser for $17.50. Nancy buts 3 pens, 3 pencils, and 2 erasers for $ 10. James buys 5 pens, 2 pencils, and 5 erasers for $14.50. How much will Simon have to pay if he buys 3 pens, 2 pencils, and 5 erasers?


Note: How often in the GMAT do such questions appear?
Can i have a generic way to solve such questions?

[ganeshrkamath]

Four friends Laura, Nancy, James and Simon buy some pencils, pens and erasers from a shop. Laura buys 5 pencils, 6 pens, and 1 eraser for $17.50. Nancy buts 3 pens, 3 pencils, and 2 erasers for $ 10. James buys 5 pens, 2 pencils, and 5 erasers for $14.50. How much will Simon have to pay if he buys 3 pens, 2 pencils, and 5 erasers?


Note: How often in the GMAT do such questions appear?
Can i have a generic way to solve such questions?

x = pencil
y = pen
e = eraser
17.5 = 5x + 6y + 1e________________(1)
10 = 3x + 3y + 2e _________________(2)
14.5 = 2x + 5y + 5e________________(3)

2*(1) - (2):
35 - 10 = 7x + 9y
25 = 7x + 9y_________________(4)

5*(1) - (3):
87.5-14.5 = 23x + 25y
73 = 23x + 25y_______________(5)

(4)*25 - (5)*9
625 - 657 = 175y - 207x
32 = 32x
x = 1

Substitute x in (4)
25 = 7 + 9y
9y = 18
y = 2

Substitute in x and y in (2):
10 = 3 + 6 + 2e
2e = 1
e = 1/2

Now, to buy 3 pens, 2 pencils and 5 erasers, Simon will have to pay
2x + 3y + 5e = 2 + 6 + 2.5
[spoiler]= $10.5[/spoiler]

Generic method:
1. Solve 2 equations to eliminate one variable.
2. Solve another 2 equatios to eliminate the same variable.
You have now reduced the problem to 2 equations with 2 unknowns.
3. Solve these two equations to get the 2 unknowns.
4. Substitute the unknowns in the original equation to get the third unknown.
I don't know of any other short cut. Suggestions are welcome.

Cheers

[Matt@VeritasPrep]

This isn't a very artfully written question: usually these sort of questions (which do appear on the GMAT) have some sort of clever way to combine the equations to solve for the requested variable(s). This one doesn't appear to have a particularly neat solution, but I may be missing it. As far as I can tell this is boring isolation and simplification ... which is NOT something you'll see on the GMAT, as it's time consuming and mindless.

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Hi vinay1983,

I agree with Matt that this question isn't written in "GMAT style." This question is built around a 3-variable "system" of equations, which is a concept that some Test Takers do see (especially if they're doing really well in the Quant section). However, the clunky wording and lack of a "math shortcut" makes this a poor example of the concept.

GMAT assassins aren't born, they're made,
Rich