Objective:

To use linear combination to solve systems of linear equations.

Notes:

Steps to solve a system of equations by linear combination.
  1. Think about the problem.
  2. Pick variables and write 'let' statements.
  3. Write two equations.
  4. Solve using a method from sections 7.1, 7.2, or 7.3.
  5. Write the answer in a complete sentence.

Examples:

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

    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).
    2. Pick variables and write 'let' statements.
      Let n = the number of nickels
           q = the number of quarters
    3. Write two equations.
      n + q = 52 (The number of nickels plus the number of quarters is equal to 52.) 5n + 25q = 400 (The value of the nickels plus the value of the quarters is equal to 400 cents)
      (Hint: Make sure to use the same units for all variables (pick either $ or ¢).
    4. Solve.
      I chose to use the linear combination method (7.3) for this problem. Ex 1
    5. Write your answer as a complete sentence.
      Homer has 45 nickels and 7 quarters. (doh!)



  2. Lisa has a large rectangular bedroom. The length of her bedroom is five feet more than the width. The perimeter is 70 feet. What are the dimensions of her bedroom?

    1. Think about the problem.
      Perimeter is the distance around. (P = 2l + 2w)
    2. Pick variables and write 'let' statements.
      Let l = the length of the bedroom
           w = the width of the bedroom
    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.)
    4. Solve.
      Since l is by itself already, I chose to use the substitution method (7.2) this time. Ex 2
    5. Write your answer as a complete sentence.
      Lisa's bedroom is 15ft by 20ft.


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This page was last updated on 08/09/13.