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Example 2: Solve the following system of equations:

4y - 5=20 - 3x  
4x - 7y + 16=0  

  1. Rearrange each equation:
    3x + 4y = 25
    4x - 7y = - 16
  2. Multiply the first equation by 4 and the second equation by -3:
    12x + 16y = 100
    -12x + 21y = 48
  3. Add the equations:
    37y = 148
  4. Solve for the variable:
    y = 4
  5. Plug y = 4 into one of the equations and solve for x:
    3x + 4(4) = 25
    3x + 16 = 25
    3x = 9
    x = 3
    Thus, the solution to the system of equations is (3, 4).
  6. Check:
    4(4) - 5 = 20 - 3(3) ? Yes.
    4(3) - 7(4) = - 16 ? Yes.

Example 3: Solve the following system of equations:

2x - 5y=15  
10y=20 + 4x  

  1. Rearrange each equation:
    2x - 5y = 15
    -4x + 10y = 20
  2. Multiply the first equation by 2:
    4x - 10y = 30
    -4x + 10y = 20
  3. Add the equations:
    0 = 50
Since 0≠50, this system of equations has no solutions. It is inconsistent (and independent). The equations describe two parallel lines.

Example 4: Solve the following system of equations:


6x + 14y=16  
24 - 9x=21y  

  1. Rearrange each equation:
    6x + 14y = 16
    -9x - 21y = - 24
  2. Multiply the first equation by 3 and the second equation by 2:
    18x + 42y = 48
    -18x - 42y = - 48
  3. Add the equations:
    0 = 0
Since 0 = 0 for any value of x, the system of equations has infinitely many solutions. Every ordered pair (x, y) which satisfies 6x + 14y = 16 (or -9x - 21y = - 24) is a solution to the system. The system is dependent (and consistent). The two equations describe the same line.

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