A rectangular auditorium seats 1702
people. The number of seats in each row exceeds the number of rows by 9
. Find the number of seats in each row.
Let's denote the number of seats in each row as "x" and the number of rows as "y".
According to the given information, we can set up the following equations:
x * y = 1702 (equation 1)
x = y + 9 (equation 2)
To solve this system of equations, we can substitute equation 2 into equation 1:
(y + 9) * y = 1702
Expanding the equation:
y^2 + 9y = 1702
Rearranging the equation:
y^2 + 9y - 1702 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula. However, since the coefficients are not easily factorable, let's use the quadratic formula:
y = (-b ± sqrt(b^2 - 4ac))/(2a)
Using the quadratic formula with a = 1, b = 9, and c = -1702:
y = (-9 ± sqrt(9^2 - 4*1*(-1702)))/(2*1)
y = (-9 ± sqrt(81 + 6808))/2
y = (-9 ± sqrt(6889))/2
y = (-9 ± 83)/2
For y to be a positive integer, we can discard the negative solution:
y = (-9 + 83)/2
y = 74/2
y = 37
Now that we have the value for y, we can substitute it back into equation 2 to find x:
x = y + 9
x = 37 + 9
x = 46
Therefore, the number of seats in each row is 46.