Solve the 3-variable system for x, y, z. Many won’t get RIGHT! 

Nice job on this one quicker and to the point.

x = -2, y = 5 & z = 3

First off... ? three unknowns ? requires three equations and we've got them. What are we hoping to find? (-1-) values of x, y, and z that satisfy all three equations. (-2-) Every instance of the three variabled is to the first power; i.e. no x^2 or x^3, etc. meaning exactly ONE value for each. (-3-) these are all straight line equations in 3-D space so the answer is where all three meet one another at exactly the same (x, y, z) position. eq.1 x+y-z = 0 eq.2 3x-y+3z = -2 eq.3 x+2y-3z = -1 we've established that the number of unknowns requires that many equations. By adding eq.2 to eq.3 we eliminate z. thus: 3x - y + 3z = -2 x + 2y - 3z = -1 ______________________ 4x + y = -3 eq.4 next add 3×eq.1 and add to eq.2 3x + 3y - 3z = 0 3x - y + 3z = -2 ________________________ 6x + 2y = -2 eq.5 now we have two equations in two unknowns: 4x + y = -3 eq.4 6x + 2y = -2 eq.5 multiply eq.4 × -2 => -8x -2y = 6 add eq.5 -8x - 2y = 6 6x + 2y = -2 _________________ -2x = 4 x = -2 sol.1 using either eq.4 or eq.5 to solve for y given x = -2 4x + y = -3 eq.4 4(-2) + y = -3 -8 + y = -3 y = 5 sol.2 verify in eq.5 6x + 2y = -2 eq.5 6(-2) +2(5) =? -2 -12 + 10 =❤ -2✔️ Given x = -2 sol.1 y = 5 sol.2 use eq.1 or eq.2 or eq.3 to solve for sol.3 eq.1 x+y-z = 0 -2 + 5 - z = 0 3 = z sol.3 VERIFY: sol.1 x = -2 sol.2 y = 5 sol.3 z = 3 eq.1 x+y-z = 0 -2 + 5 - 3 =? 0 3 - 3 =❤ 0✔️ eq.2 3x-y+3z = -2 3(-2) - 5 + 3(3) =? -2 -6 -5 + 9 =? -2 -11 + 9 =❤ -2✔️ eq.3 x+2y-3z = -1 -2 + 2(5) - 3(3) =? -1 -2 +10 -9 =? -1 8 - 9 =❤ -1✔️

Good stuff

Linear Algebra: See Strang's book. (4): Add 3 times Eq. 1 to Eq. 2: 6x + 2y = -2 or 3x + y = -1 (5): Subtract 3 times Eq. 1 from Eq. 3: -2x - y = -1 or 2x + y = 1 (6): Subtract Eq. 5 from Eq. 4: x = -2 (7): From Eq. 5: y = 1 - 2x = 1 - 2(-2) = 5 (8): From Eq. 1: z = x + y = -2 + 5 = 3

-2 5 -3

wow, long & involved, but after all the subs, i got it. thanks for a good one.

x = 8/3 y = 13/3

Muito prolixo!

x=-2. y=5 z=3

x +y = z and substitute for z

Ok only those who don't visit the John will get it right