There were originally 80lb in the first box and 75lb in the second box.
The first equation describes the original weight in each box.
x+y=155
The second equation describes the weight after the 20 pounds is removed from the first box and added to the second:
![x-20=\frac{12}{19}(y+20)](https://tex.z-dn.net/?f=x-20%3D%5Cfrac%7B12%7D%7B19%7D%28y%2B20%29)
, because 20 pounds is taken from the first box, x, and added to the second box, y; and that makes the first box equal to 12/19 of the second one.
We now have this system of equations:
![\left \{ {{x+y=155} \atop {x-20=\frac{12}{19}(y+20)}} \right.](https://tex.z-dn.net/?f=%20%5Cleft%20%5C%7B%20%7B%7Bx%2By%3D155%7D%20%5Catop%20%7Bx-20%3D%5Cfrac%7B12%7D%7B19%7D%28y%2B20%29%7D%7D%20%5Cright.%20)
We will simplify the bottom equation first by multiplying both sides by 19 to cancel the fraction:
![19(x-20)=19(\frac{12}{19})(y+20)](https://tex.z-dn.net/?f=19%28x-20%29%3D19%28%5Cfrac%7B12%7D%7B19%7D%29%28y%2B20%29)
We use the distributive property on the left, and cancel the 19 on the right:
![19*x-19*20=12(y+20) \\19x-380=12*y+12*20 \\19x-380=12y+240](https://tex.z-dn.net/?f=19%2Ax-19%2A20%3D12%28y%2B20%29%0A%5C%5C19x-380%3D12%2Ay%2B12%2A20%0A%5C%5C19x-380%3D12y%2B240)
We will move the 12y to the left side of the equation by subtracting:
19x-380-12y=12y+240-12y
19x-12y-380=240
Now we will cancel the 380 by adding:
19x-12y-380+380=240+380
19x-12y=620
Now our system looks like this:
![\left \{ {{x+y=155} \atop {19x-12y=620}} \right.](https://tex.z-dn.net/?f=%20%5Cleft%20%5C%7B%20%7B%7Bx%2By%3D155%7D%20%5Catop%20%7B19x-12y%3D620%7D%7D%20%5Cright.%20)
To eliminate a variable, we want the coefficients to be the same. We will multiply the top equation by 19 to achieve this:
![\left \{ {{19(x+y=155)} \atop {19x-12y=620}} \right. \\ \\ \left \{ {{19x+19y=2945} \atop {19x-12y=620}} \right.](https://tex.z-dn.net/?f=%20%5Cleft%20%5C%7B%20%7B%7B19%28x%2By%3D155%29%7D%20%5Catop%20%7B19x-12y%3D620%7D%7D%20%5Cright.%0A%5C%5C%0A%5C%5C%20%5Cleft%20%5C%7B%20%7B%7B19x%2B19y%3D2945%7D%20%5Catop%20%7B19x-12y%3D620%7D%7D%20%5Cright.%20%20)
Now we will subtract the bottom equation to cancel x:
![\left \{ {{19x+19y=2945} \atop {-(19x-12y=620)}} \right. \\ \\19y--12y=2945-620 \\19y+12y=2325 \\31y=2325](https://tex.z-dn.net/?f=%20%5Cleft%20%5C%7B%20%7B%7B19x%2B19y%3D2945%7D%20%5Catop%20%7B-%2819x-12y%3D620%29%7D%7D%20%5Cright.%20%0A%5C%5C%0A%5C%5C19y--12y%3D2945-620%0A%5C%5C19y%2B12y%3D2325%0A%5C%5C31y%3D2325)
Divide both sides by 31:
31y/31=2325/31
y=75
Substituting this back into our original first equation:
x+75=155
Subtract both sides by 75:
x+75-75=155-75
x=80