Question

В зале несколько скамеек.Если 2 ученика сядет на одну скамейку, то будут без места 7 учеников.А если на одну скамейку сядет 3 человека,5 скамеек останутся свободными.Сколько учеников и скамеек?

Answer 1

Х - количество скамеек
2х+7 - это, если каждую скамейку сядут 2 ученика и 7 учеников останутся без места
3(х-5) - это, если каждую скамейку сядут 3 ученика и 5 скамеек останутся свободными
2х+7=3(х-5)
2х+7=3х-15
3х-2х=15+7
х=22 - количество скамеек в зале
2*22+7=51 - число учеников

из 2-ого уравнения:
3(22-5)=51 - число учеников

Answer 2

Problem Analysis

We are given the following information: - If 2 students sit on one bench, 7 students will be left without a seat. - If 3 students sit on one bench, 5 benches will be left empty.

We need to determine the number of students and the number of benches.

Solution

Let's assume there are x benches and y students.

From the given information, we can form the following equations:

Equation 1: If 2 students sit on one bench, 7 students will be left without a seat. - Number of students sitting on benches: 2x - Number of students left without a seat: y - 2x - According to the given information, y - 2x = 7

Equation 2: If 3 students sit on one bench, 5 benches will be left empty. - Number of benches occupied by students: y/3 - Number of empty benches: x - (y/3) - According to the given information, x - (y/3) = 5

We can solve these two equations to find the values of x and y.

Solving the Equations

Let's solve the equations using substitution or elimination method.

From Equation 1, we have y - 2x = 7. We can rewrite this equation as y = 7 + 2x.

Substituting this value of y in Equation 2, we get x - (7 + 2x)/3 = 5.

Simplifying the equation, we have 3x - (7 + 2x) = 15.

Simplifying further, we get 3x - 7 - 2x = 15.

Combining like terms, we have x - 7 = 15.

Adding 7 to both sides, we get x = 22.

Substituting this value of x in Equation 1, we have y - 2(22) = 7.

Simplifying, we get y - 44 = 7.

Adding 44 to both sides, we get y = 51.

Answer

Therefore, there are 51 students and 22 benches in the hall.

Please note that the above solution is based on the given information and the equations formed from it.