I need help with thuis qustion. I have been working on this for three days now and I can't figure it out.?

Question

Solving Systems of Linear Equations in Four Variables.

The directions say "Solve the equations. Describe what you are doing at each step in your solution process."
2w-x+5y+z=-3
3w+2x+2y-6z=-32
w+3x+3y-z=-47
5w-2x-3y+3z=49
Please help!!!!!!!!!!!

Answer 1

That's a nasty one. :(

If you can't use matrices, it will take a lot of steps.

First, you'd have to solve one equation for one variable.

Then you'd have to plug that variable in all the other three equations and group them back into three equations in three unknowns.

Then you'd have to solve one of THOSE equations for one variable, and plug that mess in the other two equations and simplify.

Then you can solve 2 equations pretty easily, so you'd just use those 2 variables to figure out the other 2.

I'm willing to work it out with you but it would take me a while.

I'll solve it on my calculator so at least you have answers...

My calculator says the answers should be:

w = 2, x = -12, y = -4, z = 1

Answer 2

For one thing, this is HUGE. Do you know matrices? If you do, you want to reduce this matrix to reduced row echelon form.

[ 2 -1 5 1 | -3]
[3 2 2 -6 | -32]
[1 3 3 -1 | -47]
[5 -2 -3 3 | 49]

That's as far as I go.. 4 equations and 4 unknowns has the same process as 3 equations and 3 unknowns... use substitution in one of the equations, plug them into the other three to get a "3 equations, 3 unknowns" problem. Take one of these 3 equations and use substitution on the other two, giving you a "2 equations, 2 unknowns" system of equations. At this point, it should be trivial to solve for two variables.

Once you solve for the two variables, plug these values into one of your "three equations, three unknowns" equations, to get the third.

Get the fourth by plugging the three you solved for in ANY of the four equations, four unknowns.

To start you off, I'm going to solve for z in the first equation.

2w - x + 5y + z = -3
z = -2w + x - 5y - 3

Plug this into the other three equations.

3w + 2x + 2y - 6(-2w + x - 5y - 3) = -32
15w - 4x + 32y = -50

w + 3x + 3y - (-2w + x - 5y - 3) = 47
3w + 2x + 8y = 44

5w - 2x - 3y + 3(-2w + x - 5y - 3) = 49
-w + x -18y = 58

Three equations, three unknowns are:
15w - 4x + 32y = -50
3w + 2x + 8y = 44
-w + x -18y = 58

Repeat the process to get two equations, two unknowns.

Answer 3

Since you apparently can't use matrices, you ned to do this by elimination. This means you take the four equations and reduce it to 3 equations in 3 variables, take those 3 and get 2 equations in 2 variables, take those two, ane reduce it to a single equation in one variable. Then you go backwards up the chain, solving each in sequence.

Your equations (I've designated each with a letter, A, B, C, etc.):

A) 2w-x+5y+z=-3
B) 3w+2x+2y-6z=-32
C) w+3x+3y-z=-47
D) 5w-2x-3y+3z=49

Start by eliminating z. You can pick any pair of equations to do this. Notice that A has just +z while B has -6z. So 6A + B will eliminate z:

E) 6A + B -> 6(2w-x+5y+z) + (3w+2x+2y-6z)= 6(-3) + -32
E) 15w - 4x + 32y = -50

Doing this with A and C:

F) A + C = (2w-x+5y+z) + (w+3x+3y-z) = -3 + -47
F) 3w + 2x + 8y = -50

Now with B and D:

G) B + 2D = (3w+2x+2y-6z) + 2(5w-2x-3y+3z) = -32 + 2(49)
G) 13w -2x - 4y = 66

Now there are 3 (E, F, and G) equations in 3 unknowns (w, x, y). Use these to get 2 equations in w and x. First E and F:

H) E - 4F = (15w - 4x + 32y) - 4(3w + 2x + 8y) = -50 - 4(-50)
H) 3w - 12x = 150

And F and G:

J) F + 2G = (3w + 2x + 8y) + 2(13w -2x - 4y)= -50 + 2(66)
J) 29w - 2x = 82

Now its just two equations: H and J. Eliminate x:

K) H - 6J = (3w - 12x) - 6(29w - 2x) = 150 - 6(82)
K) -171w = -342
w = 2

Now you substitute back through any of the above equations one by one. Combining w=2 with J:

J) 29w - 2x = 82
J) 29(2) - 2x = 82
x = (58 - 82)/2
x = -12

Now knowing x and w, use equation G:

G) 13w -2x - 4y = 66
G) 13(2) - 2(-12) - 4y = 66
y = -4

Finally, use one of the original equations to get z:

A) 2w-x+5y+z=-3
A) 2(2) - (-12) + 5(-4) + z = -3
z = 1

It's a real pain to do all this. But the technique is pretty simple. It's just hard not to make a mistake. To check your work, pick another of the original four equations and substitute in your answers to see if it all checks.

Answer 4

Solve one of the equations for w (the third one looks pretty easy)

w = -3x - 3y + z - 47

Substitute that for w in the other three equations. Now you have three equations with three variables.

Repeat the process by solving one of the remaining three equations for x and substitute it into the other two.

Then solve one of the remaining two for y and substitute it into the final equation.

At this point you have a single equation, with a single variable (z). Solve for z, then work backwards to solve for y, x, w.