A local library has two types of study rooms: small rooms, which can accommodate up to 3 p… — SAT

A local library has two types of study rooms: small rooms, which can accommodate up to $3$ people, and large rooms, which can accommodate up to $8$ people. The library has a total of $15$ study rooms. If the maximum capacity of all the study rooms combined is $80$ people, how many large study rooms does the library have?

Cevap: 7 rooms

Cevap

The correct answer is 7.

By representing the number of small rooms as

s

and large rooms as

l

, we set up the system of linear equations

s + l = 15

and

3s + 8l = 80

. Solving the first equation for

s

gives

s = 15 - l

. Substituting this into the second equation yields

3(15 - l) + 8l = 80

. Distributing and combining like terms results in

45 + 5l = 80

. Subtracting 45 from both sides gives

5l = 35

, which simplifies to

l = 7

. Thus, there are 7 large study rooms.

Adım Adım Çözüm

Set up the system of linear equations representing the total number of rooms and their total capacity.

s + l = 15

and

3s + 8l = 80

We define

s

as the number of small study rooms and

l

as the number of large study rooms. The total number of rooms is 15, and the total capacity is 80.

Express the number of small rooms,

s

, in terms of the number of large rooms,

l

, using the first equation.

s = 15 - l

This allows for substitution into the capacity equation so that we can solve directly for the number of large study rooms.

Substitute the expression for

s

into the capacity equation and solve the resulting single-variable equation for

l

3(15 - l) + 8l = 80 \implies 45 - 3l + 8l = 80 \implies 45 + 5l = 80 \implies 5l = 35 \implies l = 7

Substituting

s

eliminates one variable, leaving a linear equation in terms of

l

that can be solved using standard algebraic steps.

Anahtar Kavram

Solving systems of linear equations in a real-world context using substitution.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru