Question 164534
"The difference of two numbers is 33" translates to {{{x-y=33}}} and the statement "four times the lesser number is subtracted from three times the greater number, the difference is 62" is the equation {{{3x-4y=62}}}






So we have the system of equations:


{{{system(x-y=33,3x-4y=62)}}}



Let's solve the system by use of substitution.



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=33}}} Start with the first equation



{{{-y=33-x}}}  Subtract {{{x}}} from both sides



{{{-y=-x+33}}} Rearrange the equation



{{{y=(-x+33)/(-1)}}} Divide both sides by {{{-1}}}



{{{y=((-1)/(-1))x+(33)/(-1)}}} Break up the fraction



{{{y=x-33}}} Reduce




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


Since {{{y=x-33}}}, we can now replace each {{{y}}} in the second equation with {{{x-33}}} to solve for {{{x}}}




{{{3x-4highlight((x-33))=62}}} Plug in {{{y=x-33}}} into the second equation. In other words, replace each {{{y}}} with {{{x-33}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{3x+(-4)(1)x+(-4)(-33)=62}}} Distribute {{{-4}}} to {{{x-33}}}



{{{3x-4x+132=62}}} Multiply



{{{-x+132=62}}} Combine like terms on the left side



{{{-x=62-132}}}Subtract 132 from both sides



{{{-x=-70}}} Combine like terms on the right side



{{{x=(-70)/(-1)}}} Divide both sides by -1 to isolate x




{{{x=70}}} Divide






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



So the first part of our answer is: {{{x=70}}}










Since we know that {{{x=70}}} we can plug it into the equation {{{y=x-33}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=x-33}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=(70)-33}}} Plug in {{{x=70}}}



{{{y=70-33}}} Multiply



{{{y=37}}} Combine like terms 




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



So the second part of our answer is: {{{y=37}}}










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


So the answers are:



{{{x=70}}} and {{{y=37}}}



This means that the two numbers are 70 and 37