Answer: The first number is a = 12 and the second number is b = 9.
Step-by-step explanation: Given that the first number a and second number b is related to each other by the following system of equations :
\(a+3b=39~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\\\\dfrac{1}{2}a+\dfrac{1}{3}b=9~~~~~~~~~~~~~~~~~~~~~~(ii)\)
We are to find the first and second number, i.e., a and b.
From equation (i), we have
\(a+3b=39\\\\\Rightarrow a=39-3b~~~~~~~~~~~~~~~~~~~~~`(iii)\)
Substituting the value of a from equation (iii) in equation (ii), we get
\(\dfrac{1}{2}\times(39-3b)+\dfrac{1}{3}b=9\\\\\\\Rightarrow \dfrac{3(39-3b)+2b}{6}=9\\\\\Rightarrow 117-9b+2b=9\times6\\\\\Rightarrow 117-7b=54\\\\\Rightarrow 7b=117-54\\\\\Rightarrow 7b=63\\\\\Rightarrow b=\dfrac{63}{7}\\\\\Rightarrow b=9.\)
Putting the value of b in equation (iii) we get
\(a=39-3\times9=39-27=12.\)
Thus, the first number is a = 12 and the second number is b = 9.
