Question

4-Step Equation
w+x+y+z=6
3w-x+y-z=-3
2w+2x-2y+z=4
2w-x-y+z=-4

**answers will be in fractions

Answer 1

This is not a "four-step equation." This is a system of four equations in four unknowns that you are to "solve simultaneously." How have you solved similar problems in the past?

Answer 2

Only three step, i kind of know how to begin but i get lost..

Answer 3

And the method? Simple elimination, substitution, Gauss Elimination, Cramer's Rule?
Most of the above methods apply as well to system of 3 variables as to 4, but it is likely to need many more steps with 4 variables. You'd need to work on this systematically, showing work on all steps. Yes, all of the values are fractions.

Answer 4

The most straightforward approach is to reduce system down to 3 equations, then reduce it down to 2 equations, finally to 1 equation. Then once you know 1 variable, use back-substitution to find all other variables.

To reduce the system, you must eliminate a variable by combining equations.
Eliminate "z" by adding equations (1) and (2). (It also gets rid of x)
\[(5) 4w +2y = 3\]
Eliminate "z" by subtracting (4) from (3). (It also gets rid of w)
\[(6) 3x - y = 8\]
Eliminate "z" by adding (2) and (4).
\[(7) 5w -2x = -7\]

Now we have system of 3 equations, 3 variables
Continue by eliminating "y". Add (5) with 2*(6)
\[(8) 4w + 6x = 19\]

Now we have system of 2 variables.
Continue by eliminating "x"
Add 3*(7) with (8)
\[19w = -2\]
\[w = -\frac{2}{19}\]
Now we can use back-substitution to find rest of variables.
plug w into equation (8), solve for x
\[4(-\frac{2}{19}) +6x = 19\]
\[x = \frac{19^2 +8}{6*19} = \frac{123}{38}\]
plug x into (6), solve for y:
\[3(\frac{123}{38}) -y = 8\]
\[y = \frac{65}{38}\]
Finally, plug w,x,y into (1) . solve for z
\[-\frac{2}{19} + \frac{123}{38} + \frac{65}{38} + z = 6\]
\[z = \frac{228+4-123-65}{38} = \frac{22}{19}\]