Solve the system

Question

x₁+x₂=-4

x₂+x₃=-4

x₃₊x₄=4

x1+x₄=4

 

x1=

x2=

x3=

x4=

Answer 1

If we subtract the second equation from the first, we get:

 

   x₁ + x₂ = -4

-(x₂ + x₃ = -4)

______________

x₁ - x₃ = 0

 

Thus, x₁ = x₃

 

Solving the first equation for x₂ now, we get:

 

x₂ = -4 - x₁

 

And solving the last equation for x₄ gives us:

 

x₄ = 4 - x₁

 

Therefore, we can express all of the variables in terms of just x₁, but x₁ is free to be whatever number we want…this is why x₁ is called a free variable. 

 

So our final answer is:

 

x₁ = x₁

x₂ = -4 - x₁

x₃ = x₁

x₄ = 4 - x₁