Question

A rectangular auditorium seats 1749 people the number of seats in each row exceed the number of rows by 20 find the number of seats in each row

Answer 1

Let x = the number of seats

Let y = the number of rows.

Since the auditorium is rectangular and it has 1,749 people, then we can say that:

`x* y=1749`

Then, if the number of seats "x" exceeds the number of rows "y" by 20, then we can say that:

`\begin{gathered} seat=row+20 \\ x=y+20 \end{gathered}`

Now we have two equations. To solve for x, let's use the substitution method.

1. Rewrite the equation 2 x = y + 20 into y = x - 20.

2. Replace the value of "y" in equation 1 by x - 20.

`\begin{gathered} xy=1749 \\ x(x-20)=1749 \end{gathered}`

2. Multiply x and x - 20.

`x^2-20x=1749`

3. Transfer the constant term 1749 on the left side of the equation. When transferring over the equal sign, the operation will change. From +1749, it becomes -1749.

`x^2-20x-1749=0`

4. To solve this quadratic equation, let's find the factors of -1749 that sums to -20.

a. 3 and -583 = -580

b. 11 and -159 = -148

c. 33 and -53 = -20

As we can see above, the factors of -1749 that sums to -20 are 33 and -53. Hence, the quadratic equation above can be factored to:

`(x+33)(x-53)=0`

5. Equate each factor to zero and solve for x.

`\begin{gathered} x+33=0 \\ x=-33 \end{gathered}`
`\begin{gathered} x-53=0 \\ x=53 \end{gathered}`

Since the value of x cannot be negative, then the value of x is 53.

Therefore, the number of seats in each row is 53. In addition, there are 33 rows in the auditorium.