Answer
$a=\frac{1}{2}$
$b=\frac{1}{3}$
Work Step by Step
$4a+18b=8$
$ \Longleftrightarrow $
$4a=8-18b$ ............................ eq (1)
$6a=7b+\frac{2}{3}$ ............. eq (2)
Multiplying eq (1) by 3 and equation (2) by 2
$12a=24-54b$ ................... eq (3)
$12a=14b+\frac{4}{3}$ ..................... eq (4)
Comparing eq (3) and eq (4)
$24-54b=14b+\frac{4}{3}$
$-54b-14b=\frac{4}{3}-24$
$-68b=-\frac{68}{3}$
$68b=\frac{68}{3}$
$b=\frac{68}{3}\frac{1}{68}=\frac{1}{3}$
Putting in eq (4)
$12a=14\times\frac{1}{3}+\frac{4}{3}$
$12a=6$
$a=\frac{6}{12}=\frac{1}{2}$
Checking
Putting $a=\frac{1}{2}$,$b=\frac{1}{3}$ in equation (1)
$4\times\frac{1}{2}+18\times\frac{1}{3}=8$
$2+6=8$
$8=8$
$LHS+RHS$
Putting $a=\frac{1}{2}$,$b=\frac{1}{3}$ in Equation (2)
$6\times\frac{1}{2}=7\times\frac{1}{3}+\frac{2}{3}$
$3=3$
$LHS=RHS$
