SOLUTION: x+y-2z=0 3x+y=1 5x+3y+7z=2 x= y= z=

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: x+y-2z=0 3x+y=1 5x+3y+7z=2 x= y= z=      Log On


   

Question 253952: x+y-2z=0
3x+y=1
5x+3y+7z=2
x=
y=
z=
Found 3 solutions by jim_thompson5910, dabanfield, drk:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
3x%2By=1 Start with the second equation.


y=-3x%2B1 Subtract 3x from both sides.


x%2By-2z=0 Move back to the first equation


x-3x%2B1-2z=0 Plug in y=-3x%2B1


-2x%2B1-2z=0 Combine like terms.


-2x-2z=-1 Subtract 1 from both sides.


2x%2B2z=1 Multiply every term by -1


So let's make 2x%2B2z=1 equation 4


5x%2B3y%2B7z=2 Move onto the third equation


5x%2B3%28-3x%2B1%29%2B7z=2 Plug in y=-3x%2B1


5x-9x%2B3%2B7z=2 Distribute


-4x-3%2B7z=2 Combine like terms.


-4x%2B7z=2-3 Subtract 3 from both sides.


-4x%2B7z=-1 Combine like terms. We'll make this equation 5.

--------------------------------------------------------------------------

So we have the new system of equations:

system%282x%2B2z=1%2C-4x%2B7z=-1%29


2%282x%2B2z%29=2%281%29 Multiply the both sides of the first equation by 2.


4x%2B4z=2 Distribute and multiply.


So we now have

system%284x%2B4z=2%2C-4x%2B7z=-1%29


Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:


%284x%2B4z%29%2B%28-4x%2B7z%29=%282%29%2B%28-1%29


%284x%2B-4x%29%2B%284z%2B7z%29=2%2B-1 Group like terms.


0x%2B11z=1 Combine like terms.


11z=1 Simplify.


z=%281%29%2F%2811%29 Divide both sides by 11 to isolate z.


------------------------------------------------------------------


4x%2B4z=2 Now go back to the first equation.


4x%2B4%281%2F11%29=2 Plug in z=1%2F11.


4x%2B4%2F11=2 Multiply.


11%284x%2B4%2Fcross%2811%29%29=11%282%29 Multiply both sides by the LCD 11 to clear any fractions.


44x%2B4=22 Distribute and multiply.


44x=22-4 Subtract 4 from both sides.


44x=18 Combine like terms on the right side.


x=%2818%29%2F%2844%29 Divide both sides by 44 to isolate x.


x=9%2F22 Reduce.


y=-3x%2B1 Go back to the previously isolated equation.


y=-3%289%2F22%29%2B1 Plug in x=9%2F22


y=-5%2F22 Combine like terms.


So the solutions are x=9%2F22, y=-5%2F22 and z=1%2F11.


Which form the ordered triple .


This means that the system is consistent and independent.

Answer by dabanfield(803) About Me  (Show Source):
You can put this solution on YOUR website!
1)x+y-2z=0
2)3x+y=1
3)5x+3y+7z=2
From equation 1 we have y=2z-x
So if (from equation 2) 3x+y = 1 then
3x+(2z-x)= 1
3x+2z-x = 1
and
From equation 3:
5x+3*(2z-x) + 7z = 2
Solve these two equations for x an z then substitute in the first equation to find y.



Answer by drk(1908) About Me  (Show Source):
You can put this solution on YOUR website!
We have three equations:
(i) x%2By-2z=0
(ii) 3x%2By=1
(iii) 5x%2B3y%2B7z=2
solve (ii) for y. We get
(iv) y+=+-3x+%2B+1
substitute (iv) into (i) and (iii) to get 2 equations with two variables.
(v) x%2B%28-3x%2B1%29-2z=0
(vi) 5x%2B3%28-3x%2B1%29%2B7z=2
simplify (v) and (vi) to get
(vii) -2x-2z=-1%7D%7D%0D%0A%28viii%29+%7B%7B%7B-4x%2B7z=-1
multiply (vii) by -2 to get
(ix) 4x+%2B+4z+=+2
(viii) -4x+%2B+7z+=+-1
add down to eliminate x and solve for z. We get
(x) 11z = 1 - -> z = 1/11.
Find x.
x = 9/11
Find y.
y = -16/22
So, we have
{9/22 , -5/22 , 1/11}