SOLUTION: The question is: Almonds worth $8.75/lb mixed with peanuts worth 4.75/lb take how many pounds of each to make an 80lb mixture worth $6.50/lb? I have tried solving by addition wi

Algebra ->  College  -> Linear Algebra -> SOLUTION: The question is: Almonds worth $8.75/lb mixed with peanuts worth 4.75/lb take how many pounds of each to make an 80lb mixture worth $6.50/lb? I have tried solving by addition wi      Log On


   

Question 83564: The question is: Almonds worth $8.75/lb mixed with peanuts worth 4.75/lb take how many pounds of each to make an 80lb mixture worth $6.50/lb?
I have tried solving by addition with 8.75a + 4.75p = 80(6.50) or $520 by mutiplying a+b=80 by -4.75 to eliminate the peanuts.The result I get is 4.75a=$520-$380 or $140. $140/4.75 is 29.47 for almonds which is wrong. I have the correct answer and my calculation doesn't match the answer and I don't know what I'm doing wrong. I need to know the right way to get to 45lbs of almonds and 35 lbs of peanuts.
This is not from a textbook but a final review.
Thank you very much
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let x=almonds, y=peanuts
Set up the following system of equations
x%2By=80
8.75x%2B4.75y=520

Multiply the 2nd equation by 100 to get whole numbers

100%288.75x%2B4.75y%29=100%28520%29

875x%2B475y=52000

So now we have the system
x%2By=80
875x%2B475y=52000

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


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=80
875%2Ax%2B475%2Ay=52000

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 1 and 875 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 875 is 875, we need to multiply both sides of the top equation by 875 and multiply both sides of the bottom equation by -1 like this:

875%2A%281%2Ax%2B1%2Ay%29=%2880%29%2A875 Multiply the top equation (both sides) by 875
-1%2A%28875%2Ax%2B475%2Ay%29=%2852000%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
875%2Ax%2B875%2Ay=70000
-875%2Ax-475%2Ay=-52000

Notice how 875 and -875 add to zero (ie 875%2B-875=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%28875%2Ax-875%2Ax%29%2B%28875%2Ay-475%2Ay%29=70000-52000

%28875-875%29%2Ax%2B%28875-475%29y=70000-52000

cross%28875%2B-875%29%2Ax%2B%28875-475%29%2Ay=70000-52000 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:

400%2Ay=18000

y=18000%2F400 Divide both sides by 400 to solve for y



y=45 Reduce


Now plug this answer into the top equation 1%2Ax%2B1%2Ay=80 to solve for x

1%2Ax%2B1%2845%29=80 Plug in y=45


1%2Ax%2B45=80 Multiply



1%2Ax=80-45 Subtract 45 from both sides

1%2Ax=35 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%2835%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=35 Multiply the terms on the right side


So our answer is

x=35, y=45

which also looks like

(35, 45)

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

1%2Ax%2B1%2Ay=80
875%2Ax%2B475%2Ay=52000

we get



graph of 1%2Ax%2B1%2Ay=80 (red) 875%2Ax%2B475%2Ay=52000 (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 (35,45). This verifies our answer.



So you need 35 pounds of almonds and 45 pounds of peanuts (note: the order is reversed check the problem or the answer you have)