Question 182960
You can solve this system with the decimal values, but I always find it easier to get rid of the decimal numbers. To do this, simply multiply both sides of the equations by 100 (which will move EVERY decimal place 2 spots to the right)



{{{1.70x + 1.30y = 37.90}}} Start with the first equation.



{{{100(1.70x+1.30y)=100(37.90)}}} Multiply both sides by 100 to clear out the decimals.



{{{170x+130y=3790}}} Distribute and multiply.



So the new first equation is {{{170x+130y=3790}}}


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{{{5.60x + 5.40y = 147.20}}} Move onto the second equation.



{{{100(5.60x+5.40y)=100(147.20)}}} Multiply both sides by 100 to clear out the decimals.



{{{560x+540y=14720}}} Distribute and multiply.



So the new second equation is {{{560x+540y=14720}}}


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So we have the system of equations:



{{{system(170x+130y=3790,560x+540y=14720)}}}



Let's use substitution to solve this system.



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


{{{170x+130y=3790}}} Start with the first equation



{{{130y=3790-170x}}}  Subtract {{{170x}}} from both sides



{{{130y=-170x+3790}}} Rearrange the equation



{{{y=(-170x+3790)/(130)}}} Divide both sides by {{{130}}}



{{{y=((-170)/(130))x+(3790)/(130)}}} Break up the fraction



{{{y=(-17/13)x+379/13}}} Reduce




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Since {{{y=(-17/13)x+379/13}}}, we can now replace each {{{y}}} in the second equation with {{{(-17/13)x+379/13}}} to solve for {{{x}}}




{{{560x+540highlight(((-17/13)x+379/13))=14720}}} Plug in {{{y=(-17/13)x+379/13}}} into the second equation. In other words, replace each {{{y}}} with {{{(-17/13)x+379/13}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{560x+(540)(-17/13)x+(540)(379/13)=14720}}} Distribute {{{540}}} to {{{(-17/13)x+379/13}}}



{{{560x-(9180/13)x+204660/13=14720}}} Multiply



{{{(13)(560x)-cross(13)((9180/cross(13))x)+cross(13)(204660/cross(13))=(13)(14720)}}} Multiply EVERY term by the LCD 13. This will eliminate the fractions 




{{{7280x-9180x+204660=191360}}} Multiply and simplify




{{{-1900x+204660=191360}}} Combine like terms on the left side



{{{-1900x=191360-204660}}} Subtract 204660 from both sides



{{{-1900x=-13300}}} Combine like terms on the right side



{{{x=(-13300)/(-1900)}}} Divide both sides by -1900 to isolate x




{{{x=7}}} Divide






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



So the first part of our answer is: {{{x=7}}}










Since we know that {{{x=7}}} we can plug it into the equation {{{y=(-17/13)x+379/13}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=(-17/13)x+379/13}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=(-17/13)(7)+379/13}}} Plug in {{{x=7}}}



{{{y=-119/13+379/13}}} Multiply



{{{y=20}}} Combine like terms and reduce.  



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



So the second part of our answer is: {{{y=20}}}










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


So our answers are:


{{{x=7}}} and {{{y=20}}}



So this means that they made 7 forks and 20 spoons