Question 162853
Let x=# of aerospace employees and y=# of agricultural employees


From the problem, we know that there are a total of 3,000 employees from both divisions. So this means that the sum of the two groups will be 3,000. This means that {{{x+y=3000}}}



Also, since 12% of the total workers will be laid off, this means that 0.12(3000)=360 employees will be laid of in total in these two divisions. So the sum of the layoffs in the aerospace division (which is 10% of "x" and is represented by 0.10x) and the layoffs in the aerospace division (which is 15% of "y" and is represented by 0.15y) will equal 360. This translates to: {{{0.10x+0.12y=360}}})



{{{0.10x+0.15y=360}}} Start with the second equation



{{{10x+15y=36000}}} Multiply EVERY term by 100 to make every number a whole number. Note: this will move the decimal point two spots to the right.



So we have the system of equations:


{{{system(x+y=3000,10x+15y=36000)}}}



{{{-10(x+y)=-10(3000)}}} Multiply the both sides of the first equation by -10.



{{{-10x-10y=-30000}}} Distribute and multiply.



So we have the new system of equations:


{{{system(-10x-10y=-30000,10x+15y=36000)}}}



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:



{{{(-10x-10y)+(10x+15y)=(-30000)+(36000)}}}



{{{(-10x+10x)+(-10y+15y)=-30000+36000}}} Group like terms.



{{{0x+5y=6000}}} Combine like terms. Notice how the x terms cancel out.



{{{5y=6000}}} Simplify.



{{{y=(6000)/(5)}}} Divide both sides by {{{5}}} to isolate {{{y}}}.



{{{y=1200}}} Reduce.



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



{{{-10x-10y=-30000}}} Now go back to the first equation.



{{{-10x-10(1200)=-30000}}} Plug in {{{y=1200}}}.



{{{-10x-12000=-30000}}} Multiply.



{{{-10x=-30000+12000}}} Add {{{12000}}} to both sides.



{{{-10x=-18000}}} Combine like terms on the right side.



{{{x=(-18000)/(-10)}}} Divide both sides by {{{-10}}} to isolate {{{x}}}.



{{{x=1800}}} Reduce.



So our answer is {{{x=1800}}} and {{{y=1200}}}.



This means that there are 1,800 aerospace employees and 1,200 agricultural employees (before the layoffs).