Ok, let us start with the problem.18x+23y+52z+69w=530x+y+z+w=20but now we have too many variables, so we eliminate one, the 'y' has a prime coefficient so that one should go first. We set the second equation equal to 'y'y=20xzwand substitute in the first.18x+23(20xzw)+52z+69w=53018x+46023x23z23w+52z+69w=530(18x23x)+(52z23z)+(69w23w)=5304605x+29z+46y=70 We still have too many variables but there is nothing we can do about it so we have to think. . . (the answer is obvious, but the reasoning goes like this) 29+46=75...5 =70! so x=1,z=1,w=1 and we can now substitute in the original equationx+y+z+w=201+y+1+1=20y+3=20y=203y=17and we corroborate our result:18(1)+23(17)+52(1)+69(1)=53018+391+52+69=53 0409+121=530530=530Q.E.D.
