Solve for all three unknowns

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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
by Level 11 User (78.4k points)