Objective:
To use linear combination to solve systems of linear equations.
Notes:
Steps to solve a system of equations by linear combination.
- Think about the problem.
- Pick variables and write 'let' statements.
- Write two equations.
- Solve using a method from sections 7.1, 7.2, or 7.3.
- Write the answer in a complete sentence.
Examples:
- 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?
- 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).
- Pick variables and write 'let' statements.
Let n = the number of nickels
q = the number of quarters
- 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 ¢).
- Solve.
I chose to use the linear combination method (7.3) for this problem.
- Write your answer as a complete sentence.
Homer has 45 nickels and 7 quarters. (doh!)
- 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?
- Think about the problem.
Perimeter is the distance around. (P = 2l + 2w)
- Pick variables and write 'let' statements.
Let l = the length of the bedroom
w = the width of the bedroom
- 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.)
- Solve.
Since l is by itself already, I chose to use the substitution method (7.2) this time.
- Write your answer as a complete sentence.
Lisa's bedroom is 15ft by 20ft.
