find the unknowns usuing either elimination or substitution

2x+6y+3z=9 , 5x-3y-5z=3 , 4x+3y+2z=3

Solve for all three unknowns

I'll number each equation that will be used later, so you

will know where they came from.

1) 2x + 6y + 3z = 9

2) 5x - 3y - 5z = 3

3) 4x + 3y + 2z = 3

The plan is to eliminate two of the unknowns so we can solve

for the value of the remaining unknown. Then we will use that

to solve for one of the other unknowns. Finally, we will use

both of those values to solve for the third unknown.

Adding equations 2 and 3 will eliminate the y term.

5x - 3y - 5z = 3

+(4x + 3y + 2z = 3)

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

9x - 3z = 6

4) 9x - 3z = 6

Multiply equation 2 by 2.

2 * (5x - 3y - 5z) = 3 * 2

5) 10x - 6y - 10z = 6

Add equation 5 to equation 1, again eliminating the y term.

2x + 6y + 3z = 9

+(10x - 6y - 10z = 6)

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

12x - 7z = 15

6) 12x - 7z = 15

Multiply equation 4 by 7, and multiply equation 6 by 3. This

will allow us to eliminate the z term when we subtract the two

resulting equations.

7 * (9x - 3z) = 6 * 7

7) 63x - 21z = 42

3 * (12x - 7z) = 15 * 3

8) 36x - 21z = 45

Subtract equation 8 from equation 7.

63x - 21z = 42

-(36x - 21z = 45)

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

27x = -3

9) 27x = -3

x = -3/27

x = -1/9 <<<<<<<<<<<<<<<<<<<<

Plug that into equation 4 and solve for z.

9x - 3z = 6

9(-1/9) - 3z = 6

-1 - 3z = 6

-3z = 7

z = -7/3 <<<<<<<<<<<<<<<<<<<<

Now, plug both of those values into equation 1 to solve for y.

2x + 6y + 3z = 9

2(-1/9) + 6y + 3(-7/3) = 9

-2/9 + 6y - 7 = 9

6y = 9 + 7 + 2/9

6y = 16 2/9 = 146/9

y = (146/9) * (1/6)

y = 146/54

y = 2 38/54

y = 2 19/27 <<<<<<<<<<<<<<<<<<<<

We aren't finished until we check these answers.

Plug all three values into one of the original equations.

Use number 2.

5x - 3y - 5z = 3

5(-1/9) - 3(2 19/27) - 5(-7/3) = 3

-5/9 - 6 19/9 + 35/3 = 3

-5/9 - 73/9 + 105/9 = 3

(-5 - 73 + 105)/9 = 3

27/9 = 3

3 = 3

That checked. Use number 3.

4x + 3y + 2z = 3

4(-1/9) + 3(2 19/27) + 2(-7/3) = 3

-4/9 + 6 19/9 - 14/3 = 3

-4/9 + 6 19/9 - 42/9 = 3

-4/9 + 73/9 - 42/9 = 3

(-4 + 73 - 42)/9 = 3

27/9 = 3

3 = 3

That checked, too. Might as well use number 1.

2x + 6y + 3z = 9

2(-1/9) + 6(2 19/27) + 3(-7/3) = 9

-2/9 + 12 38/9 - 7 = 9

12 36/9 - 7 = 9 ....... 36/9 is 4; add to the 12

12 + (4) - 7 = 9

16 - 7 = 9

9 = 9

The calculated values satisfy all three equations.

x = -1/9

y = 2 19/27

z = -7/3