SOLUTION: HELP!!!! Ive been trying to solve this most irritating word problem and I cant manage to find the 3rd equation. The Problem is .... COFFEE. A coffee manufacturer sells a 10-pou

Question 183069This question is from textbook Intermediate Algebra
: HELP!!!! Ive been trying to solve this most irritating word problem and I cant manage to find the 3rd equation. The Problem is ....
COFFEE. A coffee manufacturer sells a 10-pound package of coffee that consists of three flavors of coffee, vanilla flavored coffee costs $2/pound,Hazelnut costs $2.50/pound, and french roast cost $3/pound. The package contains the same amount of Hazelnut coffee as French roast coffee. The cost of the 10-pound package is $26. How many pounds of each type of coffee are in the package?

I labeled the equation as:
x- pounds of Vanilla flavored coffee
y- pounds of Hazelnut flavored coffee
z-pounds of French Roast flavored coffee
x + y + Z = 10
2x + 2.50y + 3z = $26
-Now if I can only manage to get the third variable Id be in good shape, BUT I just dont get it! What am I doing wrong??? This question is from textbook Intermediate Algebra

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Since the "package contains the same amount of Hazelnut coffee as French roast coffee", this means that is your third equation. Since you have 3 equations in 3 unknowns, you can find a unique solution (if there is one) to the system.

Here's the Updated Solution:

Start with the first equation.

Plug in . In other words, replace each "y" with "z"

Combine like terms. Let's call this equation 4.

---------------------------------------------------

Move onto the second equation

Multiply EVERY term by 10 to make every number a whole number.

Plug in

Combine like terms. Let's call this equation 5.

-----------------------------------------------------------

So we have the system of equations 4 and 5:

Let's solve this smaller system by substitution

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for z.

So let's isolate z in the first equation

Start with the first equation

Subtract from both sides

Rearrange the equation

Divide both sides by

Break up the fraction

Reduce

---------------------

Since , we can now replace each in the second equation with to solve for

Plug in into the second equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.

Distribute to

Multiply

Multiply both sides by the LCM of 2. This will eliminate the fractions.

Distribute and multiply the LCM to each side

Combine like terms on the left side

Subtract 550 from both sides

Combine like terms on the right side

Divide both sides by -15 to isolate x

Divide

Since we know that we can plug it into the equation (remember we previously solved for in the first equation).

Start with the equation where was previously isolated.

Plug in

Multiply

Combine like terms and reduce.

Now because we know that and , this tells us that also

========================= Answer =============================

So the solutions are

, and

which form the ordered triple (2,4,4)

This means that there are 2 lbs of Vanilla flavored coffee, 4 lbs of Hazelnut flavored coffee, and 4 lbs of French Roast flavored coffee
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