Question 334092: solve system of equation by elimination
2x-y+z=7
3x+2y-2z=-7
x-3y+4z=20 Found 2 solutions by nyc_function, Edwin McCravy:Answer by nyc_function(2741) (Show Source):
#1 2x - y + z = 7
#2 3x + 2y - 2z = -7
#3 x - 3y + 4z = 20
All three equations contain all three variables, so
1. Pick a variable to eliminate.
I'll pick x.
2. Pick two of the equations to eliminate that letter from.
I'll pick #1 & #3
3. Eliminate that variable from them
#1 2x - y + z = 7
#3 x - 3y + 4z = 20
Multiply #2 through by -2 and add them:
2x - y + z = 7
-2x + 6y - 8z = -40
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5y - 7z = -33
4. Pick either one of those equations
I'll pick #3
5. Put it together with the equation you haven't yet used.
I haven't used #2 yet, so I'll put #2 and #3 together:
#2 3x + 2y - 2z = -7
#3 x - 3y + 4z = 20
6. Eliminate that same variable from them
Multiply #3 through by -3 and add them:
3x + 2y + 2z = 9
-3x + 9y - 12z = -60
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11y - 10z = -51
7. Put the two equations with the eliminated variable
together as a system of two equations in two variables:
5y - 7z = -33
11y - 10z = -51
8. Solve them for the two variables that they contain.
Eliminate z by mutiplying the first one through by 10,
and the second one through by -7
50y - 70z = -330
-77y + 70z = 357
-----------------
-27y = 27
y = -1
Substitute -1 for y into:
5y - 7z = -33
5(-1) - 7z = -33
-5 - 7z = -33
-7z = -28
z = 4
9. Pick one of the original equations to substitute
the two variables you have found in the preceding step.
I'll pick #3
#3 x - 3y + 4z = 20
10. Substitute and solve for the remaining variable (the first one
picked in step 1.
I'll substitute y = -1 and z = 4 into #3
x - 3y + 4z = 20
x - 3(-1) + 4(4) = 20
x + 3 + 16 = 20
x + 19 = 20
x = 1
So the solution is (x,y,z) = (1,-1,4)
Edwin
