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:
[tex]x-20=\frac{12}{19}(y+20)[/tex], 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:
[tex] \left \{ {{x+y=155} \atop {x-20=\frac{12}{19}(y+20)}} \right. [/tex]
We will simplify the bottom equation first by multiplying both sides by 19 to cancel the fraction:
[tex]19(x-20)=19(\frac{12}{19})(y+20)[/tex]
We use the distributive property on the left, and cancel the 19 on the right:
[tex]19*x-19*20=12(y+20)
\\19x-380=12*y+12*20
\\19x-380=12y+240[/tex]
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:
[tex] \left \{ {{x+y=155} \atop {19x-12y=620}} \right. [/tex]
To eliminate a variable, we want the coefficients to be the same. We will multiply the top equation by 19 to achieve this:
[tex] \left \{ {{19(x+y=155)} \atop {19x-12y=620}} \right.
\\
\\ \left \{ {{19x+19y=2945} \atop {19x-12y=620}} \right. [/tex]
Now we will subtract the bottom equation to cancel x:
[tex] \left \{ {{19x+19y=2945} \atop {-(19x-12y=620)}} \right.
\\
\\19y--12y=2945-620
\\19y+12y=2325
\\31y=2325[/tex]
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
