Systems of Linear Equations in Three Variables Partial Fract

Systems of Linear Equations in Three Variables. Partial Fractions, and Systems of Nonlinear Equations in Two Variables Solve the system of linear equations algebraically: x + 2y - 7z = -4 2x + y + z = 13 3x + 9y + 36z = -33

Solution

x+2y-7z=-4 ,multiply by 2 =>2x+4y-14z=-8------------------->(i)

2x+y+z=13-------------->(ii)

3x+9y+36z=-33=> 3(x+3y+12z)=-33 =>x+3y+12z=-11,multiply by 2 =>2x+6y+24z=-22 ----------->(iii)

(ii)-(i)

2x+y+z-2x-4y+14z=13+8

=>-3y+15z=21

=>3(-y+5z)=21

=>-y+5z=7

=>y=5z-7------------------------>(iv)

(iii-ii)

2x+6y+24z-2x-y-z=-22-13

=>5y+23z=-35-------->(v)

substitute (iv) in (v)

=>(5*(5z-7))+23z=-35

=>25z-35+23z=-35

=>48z =0

=>z =0

z =0 ,y=5z-7

=>y=(5*0)-7

=> y =-7

z =0, y =-7 ,x+2y-7z=-4

=>x+(2*-7)-(7*0)=-4

=> x -14-0=-4

=>x=10

so solution is (x,y,z)=(10,-7,0)