Question 189716
Let 

x = smaller number

y = larger number




"The sum of two numbers is 99" translates to {{{x+y=99}}} and


"If 10 times the smaller is decreased by 2 times the larger, the result is 30" translates to {{{10x-2y=30}}}




Start with the given system of equations:


{{{system(x+y=99,10x-2y=30)}}}



Let's solve by substitution



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=99}}} Start with the first equation



{{{y=99-x}}}  Subtract {{{x}}} from both sides



{{{y=-x+99}}} Rearrange the equation




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


Since {{{y=-x+99}}}, we can now replace each {{{y}}} in the second equation with {{{-x+99}}} to solve for {{{x}}}




{{{10x-2highlight((-x+99))=30}}} Plug in {{{y=-x+99}}} into the second equation. In other words, replace each {{{y}}} with {{{-x+99}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{10x+(-2)(-1)x+(-2)(99)=30}}} Distribute {{{-2}}} to {{{-x+99}}}



{{{10x+2x-198=30}}} Multiply



{{{12x-198=30}}} Combine like terms on the left side



{{{12x=30+198}}}Add 198 to both sides



{{{12x=228}}} Combine like terms on the right side



{{{x=(228)/(12)}}} Divide both sides by 12 to isolate x




{{{x=19}}} Divide






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



So the first part of our answer is: {{{x=19}}}










Since we know that {{{x=19}}} we can plug it into the equation {{{y=-x+99}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=-x+99}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=-(19)+99}}} Plug in {{{x=19}}}



{{{y=-19+99}}} Multiply



{{{y=80}}} Combine like terms 




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



So the second part of our answer is: {{{y=80}}}










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


So our answers are:


{{{x=19}}} and {{{y=80}}}



This means that the smaller number is 19 and the larger number is 80