Question 133204



If "the difference of their ages is 12", then the first equation is {{{x-y=12}}}. Also, if "the sum of their ages is 50", then the second equation is {{{x+y=50}}}.




So let's solve this system by using substitution








Start with the given system of equations:


{{{system(x-y=12,x+y=50)}}}




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=12}}} Start with the first equation



{{{-y=12-x}}}  Subtract {{{x}}} from both sides



{{{-y=-x+12}}} Rearrange the equation



{{{y=(-x+12)/(-1)}}} Divide both sides by {{{-1}}}



{{{y=((-1)/(-1))x+(12)/(-1)}}} Break up the fraction



{{{y=x-12}}} Reduce




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


Since {{{y=x-12}}}, we can now replace each {{{y}}} in the second equation with {{{x-12}}} to solve for {{{x}}}




{{{x+highlight((x-12))=50}}} Plug in {{{y=x-12}}} into the first equation. In other words, replace each {{{y}}} with {{{x-12}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{2x-12=50}}} Combine like terms on the left side



{{{2x=50+12}}}Add 12 to both sides



{{{2x=62}}} Combine like terms on the right side



{{{x=(62)/(2)}}} Divide both sides by 2 to isolate x




{{{x=31}}} Divide






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



So the first part of our answer is: {{{x=31}}}










Since we know that {{{x=31}}} we can plug it into the equation {{{y=x-12}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=x-12}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=(31)-12}}} Plug in {{{x=31}}}



{{{y=31-12}}} Multiply



{{{y=19}}} Combine like terms 




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



So the second part of our answer is: {{{y=19}}}










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


So our answers are:


{{{x=31}}} and {{{y=19}}}



So the two ages are 31 and 19