Algebra 1: Solve Systems Using Elimination

Solving Systems of Equations Using Elimination

• Elimination method involves manipulating equations to eliminate one variable, making it easier to solve for the other.

Example Problems and Solutions

• Coordinates for (2):

  • System: 3x + y = 5 and x - 2y = 4
  • Solution: x-coordinate is 2.

• Coordinates for (4):

  • System: -3x + 2y = 5 and x - y = -3
  • Solution: y-coordinate is 4.

• Coordinates for (-5):

  • System: 4x - y = 9 and x - 3y = 16
  • Solution: y-coordinate is -5.

• Coordinates for (7):

  • System: x + 2y = 11 and x - y = 5
  • Solution: x-coordinate is 7.

Types of Solutions

• No Solution:

  • Example: System of equations - 6x - 2y = 2 and 9x - 3y = 1 indicates parallel lines with no intersection.

• Infinitely Many Solutions:

  • Example: System 2x - 3y = 6 and -2x + 3y = -6 represents overlapping lines, thus infinitely many solutions.

Testing Solutions

• Determine if specific points (x, y) are solutions to given systems.

• Testing (-1, -2):

  • System: x + 3y = -7 and 3x - 2y = 12
  • Result: No, this point does not satisfy the system.

• Testing (5, 2):

  • System: x + y = 7 and 2x - 8 = y
  • Result: Yes, this point meets both equations.

Solving by Elimination Method

• Example with (3, 0):

  • System: 2x + y = 6 and x - y = 3.
  • Solution found using elimination is (3, 0).

• Example with (6, 2):

  • System: 5x - 3y = 24 and 3x + 5y = 28.
  • Solution found via elimination is (6, 2).

• Example with (-2, -9):

  • System: x - 2y = 16 and -3x + y = -3.
  • Solution using any method gives (-2, -9).

• Example with (3, 1):

  • System: 2x - 5y = 1 and 3x + 2y = 11.
  • Using elimination method, the solution is (3, 1).