Application of Systems of Linear Equations

Problem:

Benjamin & Associates, a real estate developer, recently built condominiums in McCall, Idaho.  The condos were either two-bedroom units or three-bedroom units.  If the total number of bedrooms in the entire complex is 498, how many two-bedroom units are there?  How many three-bedroom units are there?

Solution:

Assign variables to the values we are looking for in the equation.

Let x be the number of two-bedroom units.
Let y be the number of three-bedroom units.

Create equations using the information given in the problem.

Since there are 199 condos built in the complex, the number of two-bedroom units plus the three bedroom units should equal the total units of 199.

x+y=199

Since there are a total of 498 bedrooms in the complex, 2x represents number of bedrooms coming from two-bedroom units, and 3x represents number of bedrooms coming from three-bedroom units,  the number of bedrooms from two-bedroom units plus the number of bedrooms from three-bedroom units should equal to the total number of bedrooms of 498.

2x+3y=498

 

Solve the system of equations.

x+y=199
2x+3y=498

 

Solve one equation for one of the variables.  Choose to solve for x in the first equation since it doesn’t have a coefficient and fractions can be avoided that way.
x+y=199
x+y-y=199-y
x=199-y
Substitute the expression into the other equation.
2x+3y=498
2(199-y)+3y=498
Solve for the other variable. 

  • Use the distributive property to remove parenthesis.
  • Combine like terms
  • Isolate the variable on one side of the equation
2(199-y)+3y=498
398-2y+3y=498
398+y=498
398+y-398=498-398
y=100

y represents the number of three-bedroom units.  There are 100 three-bedroom units.

x represents the number of two-bedroom units.  There are 199-100=99 two-bedroom units.