Systems of Equations

Example 2: Solve the following system of equations:

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:

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:

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.