Solving Systems of Equations

Elimination Method

Example 1:

1. Equations:

   • ( 2x + 3y = 8 )
   • ( 5x - 3y = -1 )

2. Steps:

   • Add the equations to eliminate ( y ):
     • ( 3y + (-3y) = 0 )
     • ( 2x + 5x = 7x )
     • ( 8 + (-1) = 7 )
   • Solve for ( x ):
     • ( 7x = 7 )
     • ( x = 1 )
   • Substitute ( x = 1 ) into the first equation:
     • ( 2(1) + 3y = 8 )
     • ( 2 + 3y = 8 )
     • ( 3y = 6 )
     • ( y = 2 )
   • Solution: ((x, y) = (1, 2))

Example 2:

1. Equations:

   • ( 2x + 5y = 19 )
   • ( x - 2y = -4 )

2. Steps:

   • Modify the second equation by multiplying by (-2):
     • ( -2(x - 2y) = 8 )
     • New equation: (-2x + 4y = 8)
   • Add the modified equation to the first equation:
     • ((2x - 2x) + (5y + 4y) = 19 + 8 )
     • ( 9y = 27 )
     • ( y = 3 )
   • Substitute ( y = 3 ) into the first equation:
     • ( 2x + 5(3) = 19 )
     • ( 2x + 15 = 19 )
     • ( 2x = 4 )
     • ( x = 2 )
   • Solution: ((x, y) = (2, 3))

Substitution Method

Example 1:

1. Equations:

   • ( y = 5 - 2x )
   • ( 4x + 3y = 13 )

2. Steps:

   • Substitute ( y = 5 - 2x ) into the second equation:
     • ( 4x + 3(5 - 2x) = 13 )
     • ( 4x + 15 - 6x = 13 )
     • Combine like terms: ( -2x + 15 = 13 )
     • ( -2x = -2 )
     • ( x = 1 )
   • Substitute ( x = 1 ) into ( y = 5 - 2x ):
     • ( y = 5 - 2(1) )
     • ( y = 3 )
   • Solution: ((x, y) = (1, 3))

Example 2:

1. Equations:

   • ( y = 3x + 2 )
   • ( y = 7x - 6 )

2. Steps:

   • Set the equations equal to each other:
     • ( 3x + 2 = 7x - 6 )
     • Solve for ( x ):
       • Add 6 to both sides and subtract 3x: ( 8 = 4x )
       • ( x = 2 )
   • Substitute ( x = 2 ) into ( y = 3x + 2 ):
     • ( y = 3(2) + 2 = 8 )
   • Solution: ((x, y) = (2, 8))

Example 3:

1. Equations:

   • ( 4x + 2y = 14 )
   • ( 3x - 5y = -22 )

2. Steps:

   • Solve for ( y ) in the first equation:
     • ( 2y = -4x + 14 )
     • ( y = -2x + 7 )
   • Substitute into the second equation:
     • ( 3x - 5(-2x + 7) = -22 )
     • Simplify: ( 3x + 10x - 35 = -22 )
     • ( 13x = 13 )
     • ( x = 1 )
   • Substitute ( x = 1 ) into ( y = -2x + 7 ):
     • ( y = 5 )
   • Solution: ((x, y) = (1, 5))