7.4  Problem Solving Using Linear Systems

Mrs. Agriesti's Algebra

Goal: To use linear combination to solve systems of linear equations.

Example 1:

Derrick has a jar of nickels and quarters with a value of four dollars.  There are 52 coins in all.  How many nickels does Derrick have?  How many quarters does he have?

Step 1:  Think about the problem.  There are 52 coins (nickels and quarters).  Each nickel is worth 5 cents and each quarter is worth 25 cents.  The total value is $4 (400 cents).

Step 2:  Pick variables and write "let" statements. 
       Let n = the number of nickels
             q = the number of quarters

Step 3:  Write two equations.
       n +     q =   52   (The number of nickels plus the number of quarters is equal to 52.)
     5n + 25q = 400   (Each nickel is worth 5¢ and each quarter worth 25¢, total is 400¢)
                                (Make sure to use the same units for all variables (pick either $ or ¢).

Step 4:  Solve.
    I chose to use the linear combination (7.3) method for this problem.

Step 5:  Write your answer as a complete sentence.
     Derrick has 45 nickels and 7 quarters.

Example 2:

Laura's parents have a large rectangular living room.  The length of their living room is five feet more than the width.  The perimeter is 70 feet.  What are the dimensions of the living room?

Step 1:  Think about the problem.  Perimeter is the distance around.  (P = 2l + 2w) 

Step 2:  Pick variables and write "let" statements. 
       Let  l  = the length of the living room
             w = the width of the living room

Step 3:  Write two equations.
        l = w + 5        (The length is five feet more than the width.)
     70 = 2l + 2w     (Use the formula for perimeter of a rectangle.)

Step 4:  Solve.
    Since l is by itself already, I chose to use the substitution (7.2) method this time. 

Step 5:  Write your answer as a complete sentence.
     Laura's living room is 15ft by 20ft.