For a system of linear equations to have no solution the equations of the line must be parallel to each other but not lie on the same lines i.e. the slopes of the lines must be the same but the y-intercepts must be different.
<span>We can safely assume that 1212 is a misprint and the number of seats in a row exceeds the number of rows by 12.
Let r = # of rows and s = # of seats in a row.
Then, the total # of seats is T = r x s = r x ( r + 12), since s is 12 more than the # of rows.
Then
r x (r + 12) = 1564
or
r**2 + 12*r - 1564 = 0, which is a quadratic equation.
The general solution of a quadratic equation is:
x = (-b +or- square-root( b**2 - 4ac))/2a
In our case, a = 1, b = +12 and c = -1564, so
x = (-12 +or- square-root( 12*12 - 4*1*(-1564) ) ) / 2*1
= (-12 +or- square-root( 144 + 6256 ) ) / 2
= (-12 +or- square-root( 6400 ) ) / 2
= (-12 +or- 80) / 2
= 34 or - 46
We ignore -46 since negative rows are not possible, and have:
rows = 34
and
seats per row = 34 + 12 = 46
as a check 34 x 46 = 1564 = total seats</span>