Solve a system without a unique solution

Learn how to solve a system without a unique solution or a system with infinitely many solutions. 

Example #1

Solve the following system by elimination

1. 4x - y = 5

2.-4x + y = -5

Add the left sides and the right sides of 1. and 2.

1. 4x - y = 5

2.-4x + y = -5
_____________
    0 + 0  = 0

0 = 0

As we can see, solving this system by elimination gives an equation that is always true. Therefore, the two equations in the system represent the same line. Since the system represents the same line, the system has an infinite number of solutions. 

Here is how to write the solution:

{(x, y) \ y = 4x - 5}

Just pick anything for x, and find y.

Suppose x  = 1, then y = 4(1) - 5 = 4 - 5 = -1

One solution is (1, -1)

Check

1. 4(1) - -1 = 5

2.-4(1) + -1 = -5

1. 4 + 1 = 5

2.-4 + -1 = -5

1. 5 = 5

2.-5 = -5

Just keep choosing random values for x to find more solutions. For example, the following 5 pairs are all solutions to the system of equation.

{(0,-5),(2,3),(3,7),(-1,-9),(-2,-13)} 

Example #2

Solve the following system by elimination

1. 15x - 5y = 35

2.-3x + y = -7

Multiply each side of equation 2. by 5

1. 15x - 5y = 35

2. 5(-3x + y) = -7(5)

1. 15x - 5y = 35

2. -15x + 5y = -35

Add the left sides and the right sides of 1. and 2.

1. 15x - 5y = 35

2. -15x + 5y = -35
________________
      0 + 0     = 0

0 = 0

The system has an infinite number of solutions and the solutions can be written as {(x, y) \ y = 3x - 7}