To solve this problem, we need to set up a system of equations based on the given information. Let \( r \) be the number of rows and \( s \) be the number of seats per row. We know that the total number of seats is 1316, so \( r \times s = 1316 \). Additionally, the number of seats in each row exceeds the number of rows by 19, so \( s = r + 19 \). We can substitute the second equation into the first to solve for \( r \), and then use that value to find \( s \).
Let \( r \) be the number of rows in the auditorium. According to the problem, the number of seats in each row exceeds the number of rows by 19. Therefore, the number of seats in each row can be expressed as \( r + 19 \).
The total number of seats in the auditorium is given as 1316. Therefore, we can set up the equation based on the total number of seats:
\[
r \times (r + 19) = 1316
\]
Expand the equation:
\[
r^2 + 19r = 1316
\]
Rearrange it into a standard quadratic equation form:
\[
r^2 + 19r - 1316 = 0
\]
To solve the quadratic equation \( r^2 + 19r - 1316 = 0 \), we can use the quadratic formula:
\[
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = 19 \), and \( c = -1316 \).
Calculate the discriminant:
\[
b^2 - 4ac = 19^2 - 4 \times 1 \times (-1316) = 361 + 5264 = 5625
\]
Calculate the square root of the discriminant:
\[
\sqrt{5625} = 75
\]
Substitute back into the quadratic formula:
\[
r = \frac{-19 \pm 75}{2}
\]
Calculate the two possible solutions for \( r \):
\[
r = \frac{-19 + 75}{2} = \frac{56}{2} = 28
\]
\[
r = \frac{-19 - 75}{2} = \frac{-94}{2} = -47
\]
Since the number of rows cannot be negative, we have \( r = 28 \).
The number of seats in each row is \( r + 19 \):
\[
r + 19 = 28 + 19 = 47
\]
The number of seats in each row is \(\boxed{47}\).
Answered
