solve for all variables

Question

solve by elimination method

x+2y+z=7

x-y+z=1

2x+3y-z=9

Answer 1

Hi Kes, Did you know that equations can be added or subtracted to make new equations? Also equations are still equations if you do the same thing to each side.

 

So looking at the 1st two equations I see I can subtract the 2nd one from the 1st one and get rid of x and z x+2y+z=7 minus x-y+z=1.

 

the result is 3y=6 so we divide this by 3 to get y=2.

 

So now we are down to two unknowns. We can take the last two of the original equations. plug in 2 for y.

 

x-2+z=1 2x+6-z=9 Add these equations together to get rid of z. 3x+4=10 3x=6 x=2

 

Now just plug in x and y into any of the 3 original equations. 2+4+z=7

 

z=1

 

Now plug all three solutions in all three equations for proof.

 

 

2+4+1=7

2-2+1=1

4+6-1=9

 

They all work.

Answer 2

1. Add the first and third equations to eliminate z

 

x + 2y + z = 7          Eq. 1

2x + 3y -z = 9          Eq. 3

________________

 

3x + 5y = 16            Eq. 4

 

Multiply first equation by -3 and third equation by 2, then add to eliminate y:

 

-3x - 6y - 3z = -21      Eq. 1'

4x + 6y -2z = 18        Eq. 3'

______________________

 

x -5z = -3                  Eq. 5

 

Substitute value for z from Eq. 2 (i.e. z = 1 - x + y) into Eq. 5:

 

x - 5(1 - x + y) = -3

x - 5 + 5x - 5y = -3

6x - 5y = 2                Eq. 6

 

Add equations 4 and 6

 

3x + 5y = 16

6x - 5y = 2

________________________

 

9x = 18

 

x = 2

 

But 6(x) - 5(y) = 2 ⇒ (6)*(2) - 5y = 2 ⇒ -5y = -10 ⇒ y = 2

 

Finally, z = 1 - x + y ⇒ z = 1 -2 +2 ⇒ z = 1