# Convert base y to base 10

I have a problem to find base y. The equation given is

($1111011$)gray + ($123$)y + ($211.1$)3 + ($34.4$)$6$ = (CD)$16$ + ($40$)y - ($10010$)BCD.

I am able to simplify by converting everything to base $10$ but cannot figure out how to find base y.

The simplified, converted equation becomes ($123$)y - ($40$)y = ($66$)$10$.

Any help would be grateful, thanks in advance!

## 1 Answer

Assuming your simplified, converted equation is correct, i.e. that $(123)_y-(40)_y=(66)_{10}$, we can write this as: $$1\cdot y^2+2\cdot y+3-(4\cdot y+0)=6\cdot 10+6$$ which we can simplify to: $$y^2-2y-63=0$$ Solving for $y$ then yields $y=1\pm 8$. Clearly $y\neq -7$, so we get $y=9$.