Question

An apartment company is planning to build an additional set of units at an existing apartment complex. There will be a total of 48 new apartment units that are either single-bedroom units or double-bedroom units. Since double-bedroom units are easier to rent, the company wants to have twice as many double-bedroom units as single-bedroom units.

Set up the two equations that can be used to find the number of single-bedroom units, s, and double-bedroom units, d, that the company is planning to build. Let the equation that represents the total number of units be referred to as constraint 1, and let the equation that represents the comparison number of units be referred to as constraint 2.

Determine if the number of single-bedroom apartments and double-bedroom apartments listed below would meet one of the constraints, both constraints, or neither constraint for this situation.

Which constraint or constraints, if any, would be met if 15 single-bedroom units and 30 double-bedroom units were built?



Which constraint or constraints, if any, would be met if 12 single-bedroom units and 36 double-bedroom units were built?



Which values below, in the form of (s, d), represent the solution to the system of equations that represents this situation?

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Answer 1

Part 1)

`s+d=48` -----> constraint 1

`d=2s` ----> constraint 2

Part 2) see the procedure

Part 3) see the procedure

Part 4) The solution is (16,32)

Explanation:

Part 1) Set up the two equations that can be used to find the number of single-bedroom units, s, and double-bedroom units, d, that the company is planning to build.

Let

s -----> the number of single-bedroom units

d ----> the number of double-bedroom units

we know that

There will be a total of 48 new apartment units

so

`s+d=48` -----> constraint 1

The company wants to have twice as many double-bedroom units as single-bedroom units.

so

`d=2s` ----> constraint 2

Part 2) Which constraint or constraints, if any, would be met if 15 single-bedroom units and 30 double-bedroom units were built?

we have

s=15 units

d=30 units

substitute the value of s and the value of d in each constraint

Verify constraint 1

`15+30=48`

`45=48` ----> is not true

so

Constraint 1 is not met

Verify constraint 2

`d=2s`

`30=2(15)` ----> is true

so

Constraint 2 is met

Part 3) Which constraint or constraints, if any, would be met if 12 single-bedroom units and 36 double-bedroom units were built?

we have

s=12 units

d=36 units

substitute the value of s and the value of d in each constraint

Verify constraint 1

`12+36=48`

`48=48` ----> is true

so

Constraint 1 is met

Verify constraint 2

`d=2s`

`36=2(12)` ----> is not true

so

Constraint 2 is not met

Part 4) Which values below, in the form of (s, d), represent the solution to the system of equations that represents this situation

Solve the system of equations by substitution

substitute equation 2 in equation 1

`s+2s=48`

solve for s

`3s=48\\s=16`

Find the value of d

`d=2(16)=32`

therefore

The solution is (16,32)