SOLUTION: Cashews costing $5.50 per pound are mixed with pecans costing $3.00 per pound. How many pounds of each should be mixed to obtain a 20 pound cashew pecan mixture costing $3.50 per

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Question 85077: Cashews costing $5.50 per pound are mixed with pecans costing $3.00 per pound. How many pounds of each should be mixed to obtain a 20 pound cashew pecan mixture costing $3.50 per pound?
Thanks,
Tonya
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Set up the following system of equations
5.5x%2B3y=20%2A3.50
x%2By=20

Multiply

5.5x%2B3y=70
x%2By=20


Multiply the first equation by 10 to remove the decimal
55x%2B30y=700
x%2By=20

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

55%2Ax%2B30%2Ay=700
1%2Ax%2B1%2Ay=20

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 55 and 1 to some equal number, we could try to get them to the LCM.

Since the LCM of 55 and 1 is 55, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -55 like this:

1%2A%2855%2Ax%2B30%2Ay%29=%28700%29%2A1 Multiply the top equation (both sides) by 1
-55%2A%281%2Ax%2B1%2Ay%29=%2820%29%2A-55 Multiply the bottom equation (both sides) by -55


So after multiplying we get this:
55%2Ax%2B30%2Ay=700
-55%2Ax-55%2Ay=-1100

Notice how 55 and -55 add to zero (ie 55%2B-55=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2855%2Ax-55%2Ax%29%2B%2830%2Ay-55%2Ay%29=700-1100

%2855-55%29%2Ax%2B%2830-55%29y=700-1100

cross%2855%2B-55%29%2Ax%2B%2830-55%29%2Ay=700-1100 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

-25%2Ay=-400

y=-400%2F-25 Divide both sides by -25 to solve for y



y=16 Reduce


Now plug this answer into the top equation 55%2Ax%2B30%2Ay=700 to solve for x

55%2Ax%2B30%2816%29=700 Plug in y=16


55%2Ax%2B480=700 Multiply



55%2Ax=700-480 Subtract 480 from both sides

55%2Ax=220 Combine the terms on the right side

cross%28%281%2F55%29%2855%29%29%2Ax=%28220%29%281%2F55%29 Multiply both sides by 1%2F55. This will cancel out 55 on the left side.


x=4 Multiply the terms on the right side


So our answer is

x=4, y=16

which also looks like

(4, 16)

Notice if we graph the equations (if you need help with graphing, check out this solver)

55%2Ax%2B30%2Ay=700
1%2Ax%2B1%2Ay=20

we get



graph of 55%2Ax%2B30%2Ay=700 (red) 1%2Ax%2B1%2Ay=20 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (4,16). This verifies our answer.



so you need 4 pounds of cashews and 16 pounds of pecans to make a 20 pound mix costing $3.50 per pound.