Question 189685
Let 

x = amount of $9 nuts

y = amount of $6.5 nuts



Since we "to make 10 kg of nuts", this gives the first equation {{{x+y=10}}}.



Also, since "$9 per kg are to be mixed with nuts costing $6.5 per kg to make 10 kg of nuts costing $8 per kg.", this means that {{{9x+6.5y=8(10)}}}. Multiply 8 and 10 to get 80. So the equation then becomes {{{9x+6.5y=80}}}. Finally, multiply EVERY term by 10 to make every number a whole number. So the equation then becomes {{{90x+65y=800}}}




So we have the system of equations:


{{{system(x+y=10,90x+65y=800)}}}



Let's solve 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 y.





So let's isolate y in the first equation


{{{x+y=10}}} Start with the first equation



{{{y=10-x}}}  Subtract {{{x}}} from both sides



{{{y=-x+10}}} Rearrange the equation



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


Since {{{y=-x+10}}}, we can now replace each {{{y}}} in the second equation with {{{-x+10}}} to solve for {{{x}}}




{{{90x+65highlight((-x+10))=800}}} Plug in {{{y=-x+10}}} into the second equation. In other words, replace each {{{y}}} with {{{-x+10}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{90x+(65)(-1)x+(65)(10)=800}}} Distribute {{{65}}} to {{{-x+10}}}



{{{90x-65x+650=800}}} Multiply



{{{25x+650=800}}} Combine like terms on the left side



{{{25x=800-650}}}Subtract 650 from both sides



{{{25x=150}}} Combine like terms on the right side



{{{x=(150)/(25)}}} Divide both sides by 25 to isolate x




{{{x=6}}} Divide






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



So the first part of our answer is: {{{x=6}}}










Since we know that {{{x=6}}} we can plug it into the equation {{{y=-x+10}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=-x+10}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=-(6)+10}}} Plug in {{{x=6}}}



{{{y=-6+10}}} Multiply



{{{y=4}}} Combine like terms 




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



So the second part of our answer is: {{{y=4}}}










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


So our answers are:


{{{x=6}}} and {{{y=4}}}



This means that 6 kg of $9 nuts and 4 kg of $6.50 nuts are needed