Find a linear system of equations from the word problem?

[gmmstr827]

The problem:

"A commercial customer orders 81 gallons of paint that contains equal amounts of red paint, green paint, and blue paint - and hence, could be prepared by mixing 27 gallons of each. However, the store wishes to prepare this order by mixing three types of paint that are already available in large quantity:
- a reddish paint that is a mixture of 50% red, 25% green, and 25% blue paint;
- a greenish paint that is 12.5% red, 75% green, and 12.5% blue paint; and
- a bluish paint that is 20% red, 20% green, and 60% blue paint.
How many gallons of each are needed to prepare the customer's order?"

What I need/how to solve:
I need to form a linear system of equations that I can put into a matrix and then take the rref of that matrix to find the answer. If someone could supply me with those equations, that would be great.

What I've attempted:
I have identified the following:
Let x=gallons of red paint, y=gallons of green paint, and z=gallons of blue paint.
I believe that the first equation I need to use is x+y+z=81
However, from there, I'm at a loss.
I have tried using the percentages given and setting them each equal to 21, with and without the first equation I've identified, and using both whole percentage numbers and converting them to decimals. This gives no solution.

Questions:
Is x+y+z=81 one of the equations I need?
If I use the percentage amounts, do I use whole percentages or convert them to decimals?
What do I set them equal to?

I shall try setting them equal to 1 to see if that gets me anywhere (as if setting them equal to 1x, 1y, and 1z) while I wait for a response...
EDIT: Nope, 1 doesn't work. Neither does 0.

Thanks!

 

[hotvette]

Is x+y+z=81 one of the equations I need?

No, because you don't have pure red, blue, and green paint available to you. What you do have is reddish, blueish, and greenish paint. You want to mix some number of gallons of each to result in 81 gallons of paint that have equal concentrations of red, blue, and green paint.

If you want a mixture that contains 27 gallons of red paint, how much of each paint type (reddish, blueish, greenish) do you need? Since you know the concentration of red in each type you should be able to write a single equation that results in a mixture that has 27 gallons of red. Do the same for the other colors and you'll have 3 equations in 3 unknowns.

 

[gmmstr827]

Let x = gallons of reddish paint, y = gallons of bluish paint, and z = gallons of greenish paint.
I'm not sure if that definition makes sense, but the following equations do.
.5x+.125y+.2z=27
.25x+.75y+.2z=27
.25x+.125y+.6z=27
Put those values into a matrix.
[.5,.125,.2,27;.25,.75,.2,27;.25,.125,.6,27]
rref([.5,.125,.2,27;.25,.75,.2,27;.25,.125,.6,27])
= [1,0,0,40;0,1,0,16;0,0,1,25]
Therefore, there are 40 gallons of reddish paint, 16 gallons of greenish paint, and 25 gallons of bluish paint.
40+16+25=81
Therefore, the numerical answer is correct.

Are my x, y, and z definitions correct? They make sense as defining them as reddish, bluish, and greenish?

 

[hotvette]

Yep, you got it.