Question

in a theatre there are 750 seats.the number of seats in each row is 5 less than the of rows.how many seats are there in a row?

Answer 1

30 seats per row

1) Let's imagine the number of 750 seats in a row can be expressed as "s" for seats and "r" for rows. Assuming that each row has the same number of seats we can write out:

`\begin{gathered} rs=750 \\ \end{gathered}`

2) The question states also that the number of rows can be expressed as "5 less than the number of rows", i.e.

`r=s-5`

2.2) So now, let's solve this Linear System, by the Substitution Method, plugging into the second equation, the first one.

`\begin{gathered} I)rs=750 \\ II)\text{ r=s-5} \\ (s-5)s=750 \\ s^2-5s-750=0 \end{gathered}`

2.3) Solving this quadratic equation, we have (factoring the quadratic): Which two numbers whose sum is equal to -5 and which are the same numbers whose product is -750.

Sx = Sum and Px =30 x -25

`\begin{gathered} s^2-5s-750=0 \\ Sx\colon30-25=5_{}_{} \\ P_x=30\cdot-25=-75 \\ s^2-5s-750=(s+30)(s-25)_{} \\ s_1=30 \\ s_2=-25 \end{gathered}`

Now, we can state that since S =30 (just the positive number works out for that), and then there are 5 fewer seats than rows then we can write out:

r=30 -5 25 rows and 30 seats per row.

3) Hence, the answer is 30 seats per row