Hey I have a question about free variables of linear system

Hey, I have a question about free variables of linear system. Here is the question, and I have tried to solve it for many times, but I still can not get the right answer. If you can do it, please show me the process. Thank you!!!

Find the set of solutions for the linear system

2x₁ + 5x₂ - x₃ + x₄ = -18;

-x₂ - 5x₃ + 13x₄ = -5;

3x₃ - x₄ = -13

Use s1, s2, etc. for free variables if necessary

What is the answer for (x₁, x₂, x₃, x₄) = (____, _____, _____, _____)

Solution

I\'ll use the variable a,b,c,d instead of x1,x2,x3,x4

Equations are 2a+5b-c+d=-18

-b-5c+13d=-5

3c-d=-13

Adding first and third equations we get

2a+5b+2c=-31

Adding 13 times third equation and second equation we get

-b+(-5+39)c=-5-169

Therefore -b+43c=-164

Hence b-43c=164

Therefore b=164+43c

From the third equation, d=3c+13

Substituting these in the first equation, we get

2a+5(164+43c)-c+(3c+13)=-18

Therefore, 2a+217c+851=0

Hence a=(-851/2)-(217/2)c

Therefore, a=(-851/2)-(217/2)c, b=164+43c, d=3c+13

c is the free variable.

Required solution is ((-851/2)-(217/2)c, 164+43c,c,3c+13)

where c can vary