Question 198842
let x= # of pounds of soybean meal and y=# of pounds of cornmeal 


Since we want a  280 lbs mixture, this means that we have the first equation: 

{{{x+y=280}}}



Also, because we want a "280-lb mixture that is 12% protein" from the mixture of 
soybean meal that is 14% protein and cornmeal that is 7% protein, we have the second equation: {{{0.14x+0.07y=280(0.12)}}}




{{{0.14x+0.07y=280(0.12)}}} Start with the second equation.



{{{0.14x+0.07y=33.6}}} Multiply



{{{10(0.14x+0.07y)=10(33.6)}}} Multiply both sides by 100 to remove the decimals



{{{14x+7y=3360}}} Distribute 



So we have the new system of equations:


{{{system(x+y=280,14x+7y=3360)}}}



Let's solve the system by use of 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 y.





So let's isolate y in the first equation



{{{x+y=280}}} Start with the first equation



{{{y=280-x}}}  Subtract {{{x}}} from both sides



{{{y=-x+280}}} Rearrange the equation



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


Since {{{y=-x+280}}}, we can now replace each {{{y}}} in the second equation with {{{-x+280}}} to solve for {{{x}}}




{{{14x+7highlight((-x+280))=3360}}} Plug in {{{y=-x+280}}} into the second equation. In other words, replace each {{{y}}} with {{{-x+280}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{14x+(7)(-1)x + (7)(280)=3360}}} Distribute {{{7}}} to {{{-x+280}}}



{{{14x-7x+1960=3360}}} Multiply



{{{7x+1960=3360}}} Combine like terms on the left side



{{{7x=3360-1960}}}Subtract 1960 from both sides



{{{7x=1400}}} Combine like terms on the right side



{{{x=(1400)/(7)}}} Divide both sides by 7 to isolate x



{{{x=200}}} Divide




-----------------First Answer------------------------------



So the first part of our answer is: {{{x=200}}}




Since we know that {{{x=200}}} we can plug it into the equation {{{y=-x+280}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=-x+280}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=-(200)+280}}} Plug in {{{x=200}}}



{{{y=-200+280}}} Multiply



{{{y=80}}} Combine like terms 




-----------------Second Answer------------------------------



So the second part of our answer is: {{{y=80}}}







-----------------Summary------------------------------


So our answers are:


{{{x=200}}} and {{{y=80}}} which form the ordered pair (200,80) 




So 200 pounds of soybean meal and 80 pounds of cornmeal are needed for the mixture.