Question 1139578: solve by substitution
2x+3y+2z=18
-4x+y-z=5
2z=-10
Found 2 solutions by dkppathak, jim_thompson5910:
Answer by dkppathak(439)   (Show Source):
You can put this solution on YOUR website! solve by substitution
2x+3y+2z=18
-4x+y-z=5
2z=-10
solution
z=10/2=5
by substitute z=5
-4x+y=10 y=10+4x by substitution
2x+3y=8
2x+3(10+4x)=8
2x+30+12x=8
14x=8-30=-22
x=-22/14=-11/7
y=10+4x
y=10-44/7
y=70-44/7
y=26/7
answer
x=-11/7 y=26/7 z=5
Answer by jim_thompson5910(35256)  (Show Source):
You can put this solution on YOUR website!
We have these three equations
2x+3y+2z = 18
-4x+y-z = 5
2z = -10
which we'll label equation (1), equation (2), and equation (3) respectively in that exact order.
Solve equation (3) for z
2z=-10
2z/2=-10/2 .... divide both sides by 2
z = -5
Plug this into equation (2) and isolate y
-4x+y-z = 5
-4x+y-(-5) = 5 ... replace z with -5
-4x+y+5 = 5
-4x+y+5-5 = 5-5 .... subtract 5 from both sides
-4x+y = 0
-4x+y+4x = 0+4x .... add 4x to both sides
y = 4x
Call this equation (4) as we'll use it later.
Move to equation (1).
Plug in z = -5 and also y = 4x which was equation (4) we found earlier
2x+3y+2z = 18
2x+3y+2(-5) = 18 .... replace every z with -5
2x+3y-10 = 18
2x+3(4x)-10 = 18 ... replace every y with (4x), since y = 4x from equation (4)
2x+12x-10 = 18
2x+12x-10+10 = 18+10 ... add 10 to both sides
14x = 28
14x/14 = 28/14 ... divide both sides by 14
x = 2
So far we know that
x = 2
z = -5
We can plug those values into any equation that has y in it, and then solve for y. So that constitutes everything but equation (3). I'll use equation (4) since we isolated y here. It's easiest to use an equation with the variable already isolated.
y = 4*x
y = 4*2 ... replace every x with 2
y = 8
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Final Answer:
The solution as an ordered triple is (x,y,z) = (2, 8, -5)
Which breaks down to x = 2, y = 8 and z = -5
Visually, each of the three original equations plots out a plane if you're in a 3D coordinate system. Those three planes will intersect at the location (2, 8, -5). Think of 3 pieces of flat paper all meeting up at the same spot.
To verify the answer, you need to replace the variables with their proper corresponding values. Afterward, simplify the equation. You should get the same value on both sides. Getting the same value on both sides says to the reader "this equation is true". To fully verify a system of equations, all of the equations must be true for the same variable values. I'll leave the verification step up for you to try out.
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