3 Variable System of Equations Worksheets - Math Monks - Free Printable

Explanation:
Let's solve each system of equations step by step. We'll use the elimination method, which means we'll add or subtract equations to eliminate one variable at a time.

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System 1:
1) x + 3y - 3z = 12
2) 3x - y + 4z = 0
3) -x + 2y - z = 1

Step 1: Eliminate x from equations 1 and 3.
Add equation 1 and equation 3:
(x + 3y - 3z) + (-x + 2y - z) = 12 + 1
x - x = 0
3y + 2y = 5y
-3z - z = -4z
12 + 1 = 13
So: 5y - 4z = 13 → Equation A

Step 2: Eliminate x from equations 1 and 2.
Multiply equation 1 by 3:
3(x + 3y - 3z) = 3(12) → 3x + 9y - 9z = 36
Now subtract equation 2:
(3x + 9y - 9z) - (3x - y + 4z) = 36 - 0
3x - 3x = 0
9y - (-y) = 10y
-9z - 4z = -13z
36 - 0 = 36
So: 10y - 13z = 36 → Equation B

Now we have:
A) 5y - 4z = 13
B) 10y - 13z = 36

Step 3: Eliminate y. Multiply equation A by 2:
2(5y - 4z) = 2(13) → 10y - 8z = 26
Now subtract from equation B:
(10y - 13z) - (10y - 8z) = 36 - 26
10y - 10y = 0
-13z + 8z = -5z
36 - 26 = 10
So: -5z = 10 → z = -2

Step 4: Plug z = -2 into equation A:
5y - 4(-2) = 13 → 5y + 8 = 13 → 5y = 5 → y = 1

Step 5: Plug y = 1, z = -2 into equation 1:
x + 3(1) - 3(-2) = 12 → x + 3 + 6 = 12 → x + 9 = 12 → x = 3

Check in equation 2: 3(3) - 1 + 4(-2) = 9 - 1 - 8 = 0 → Correct
Check in equation 3: -3 + 2(1) - (-2) = -3 + 2 + 2 = 1 → Correct

So solution for System 1: x = 3, y = 1, z = -2

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System 2:
1) x - 6y + 4z = -12
2) x + y - 4z = 12
3) 2x + 2y + 5z = -15

Step 1: Eliminate x from equations 1 and 2.
Subtract equation 1 from equation 2:
(x + y - 4z) - (x - 6y + 4z) = 12 - (-12)
x - x = 0
y - (-6y) = 7y
-4z - 4z = -8z
12 + 12 = 24
So: 7y - 8z = 24 → Equation A

Step 2: Eliminate x from equations 1 and 3.
Multiply equation 1 by 2:
2(x - 6y + 4z) = 2(-12) → 2x - 12y + 8z = -24
Subtract from equation 3:
(2x + 2y + 5z) - (2x - 12y + 8z) = -15 - (-24)
2x - 2x = 0
2y - (-12y) = 14y
5z - 8z = -3z
-15 + 24 = 9
So: 14y - 3z = 9 → Equation B

Now we have:
A) 7y - 8z = 24
B) 14y - 3z = 9

Step 3: Eliminate y. Multiply equation A by 2:
2(7y - 8z) = 2(24) → 14y - 16z = 48
Subtract from equation B:
(14y - 3z) - (14y - 16z) = 9 - 48
14y - 14y = 0
-3z + 16z = 13z
9 - 48 = -39
So: 13z = -39 → z = -3

Step 4: Plug z = -3 into equation A:
7y - 8(-3) = 24 → 7y + 24 = 24 → 7y = 0 → y = 0

Step 5: Plug y = 0, z = -3 into equation 1:
x - 6(0) + 4(-3) = -12 → x - 12 = -12 → x = 0

Check in equation 2: 0 + 0 - 4(-3) = 12 → 12 = 12 → Correct
Check in equation 3: 2(0) + 2(0) + 5(-3) = -15 → -15 = -15 → Correct

So solution for System 2: x = 0, y = 0, z = -3

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System 3:
1) -2x - y - 4z = 6
2) -4x - 2z = -10
3) 5x + y + 6z = -1

Step 1: Eliminate y from equations 1 and 3.
Add equation 1 and equation 3:
(-2x - y - 4z) + (5x + y + 6z) = 6 + (-1)
-2x + 5x = 3x
-y + y = 0
-4z + 6z = 2z
6 - 1 = 5
So: 3x + 2z = 5 → Equation A

Step 2: Use equation 2: -4x - 2z = -10
We can simplify this by dividing by -2: 2x + z = 5 → Equation B

Now we have:
A) 3x + 2z = 5
B) 2x + z = 5

Step 3: Eliminate z. Multiply equation B by 2:
2(2x + z) = 2(5) → 4x + 2z = 10 → Equation C

Subtract equation A from equation C:
(4x + 2z) - (3x + 2z) = 10 - 5
4x - 3x = x
2z - 2z = 0
10 - 5 = 5
So: x = 5

Step 4: Plug x = 5 into equation B: 2(5) + z = 5 → 10 + z = 5 → z = -5

Step 5: Plug x = 5, z = -5 into equation 1:
-2(5) - y - 4(-5) = 6 → -10 - y + 20 = 6 → 10 - y = 6 → -y = -4 → y = 4

Check in equation 2: -4(5) - 2(-5) = -20 + 10 = -10 → Correct
Check in equation 3: 5(5) + 4 + 6(-5) = 25 + 4 - 30 = -1 → Correct

So solution for System 3: x = 5, y = 4, z = -5

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System 4:
1) -2x + 2y + z = 14
2) 3x - 2y + z = -5
3) -x + y + 2z = 3

Step 1: Eliminate z from equations 1 and 2.
Subtract equation 1 from equation 2:
(3x - 2y + z) - (-2x + 2y + z) = -5 - 14
3x + 2x = 5x
-2y - 2y = -4y
z - z = 0
-5 - 14 = -19
So: 5x - 4y = -19 → Equation A

Step 2: Eliminate z from equations 1 and 3.
Multiply equation 3 by 1: -x + y + 2z = 3
Multiply equation 1 by 2: -4x + 4y + 2z = 28
Now subtract equation 3 from this:
(-4x + 4y + 2z) - (-x + y + 2z) = 28 - 3
-4x + x = -3x
4y - y = 3y
2z - 2z = 0
28 - 3 = 25
So: -3x + 3y = 25 → Equation B

Now we have:
A) 5x - 4y = -19
B) -3x + 3y = 25

Step 3: Eliminate x. Multiply equation A by 3:
3(5x - 4y) = 3(-19) → 15x - 12y = -57
Multiply equation B by 5:
5(-3x + 3y) = 5(25) → -15x + 15y = 125

Add them:
(15x - 12y) + (-15x + 15y) = -57 + 125
15x - 15x = 0
-12y + 15y = 3y
-57 + 125 = 68
So: 3y = 68 → y = 68/3

Wait, that's not a whole number. Let's double-check.

Equation B: -3x + 3y = 25 → divide by 3: -x + y = 25/3 → y = x + 25/3

But let's recheck the subtraction step.

Equation 1: -2x + 2y + z = 14
Equation 3: -x + y + 2z = 3
Multiply equation 3 by 2: -2x + 2y + 4z = 6
Now subtract equation 1:
(-2x + 2y + 4z) - (-2x + 2y + z) = 6 - 14
-2x + 2x = 0
2y - 2y = 0
4z - z = 3z
6 - 14 = -8
So: 3z = -8 → z = -8/3

I made a mistake earlier. I should eliminate z using equations 1 and 3.

Let me start over.

Equation 1: -2x + 2y + z = 14
Equation 3: -x + y + 2z = 3

Multiply equation 1 by 2: -4x + 4y + 2z = 28
Now subtract equation 3:
(-4x + 4y + 2z) - (-x + y + 2z) = 28 - 3
-4x + x = -3x
4y - y = 3y
2z - 2z = 0
28 - 3 = 25
So: -3x + 3y = 25 → Equation B (same as before)

Equation 1 and 2:
Equation 1: -2x + 2y + z = 14
Equation 2: 3x - 2y + z = -5
Subtract equation 1 from equation 2:
(3x - 2y + z) - (-2x + 2y + z) = -5 - 14
3x + 2x = 5x
-2y - 2y = -4y
z - z = 0
-5 - 14 = -19
So: 5x - 4y = -19 → Equation A

Now solve A and B:

A) 5x - 4y = -19
B) -3x + 3y = 25

Multiply A by 3: 15x - 12y = -57
Multiply B by 4: -12x + 12y = 100

Add:
15x - 12x = 3x
-12y + 12y = 0
-57 + 100 = 43
So: 3x = 43 → x = 43/3

Then from B: -3(43/3) + 3y = 25 → -43 + 3y = 25 → 3y = 68 → y = 68/3

Then from equation 1: -2(43/3) + 2(68/3) + z = 14
-86/3 + 136/3 + z = 14
50/3 + z = 14
z = 14 - 50/3 = 42/3 - 50/3 = -8/3

So solution: x = 43/3, y = 68/3, z = -8/3

Check in equation 2: 3(43/3) - 2(68/3) + (-8/3) = 43 - 136/3 - 8/3 = 43 - 144/3 = 43 - 48 = -5 → Correct
Check in equation 3: -43/3 + 68/3 + 2(-8/3) = 25/3 - 16/3 = 9/3 = 3 → Correct

So solution for System 4: x = 43/3, y = 68/3, z = -8/3

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System 5:
1) 3x = -12 + 5y - 4z
2) 6y = 20 + 2x + 8z
3) 4x = 8z + y + 6

Step 1: Rewrite all equations in standard form (ax + by + cz = d)

Equation 1: 3x - 5y + 4z = -12
Equation 2: -2x + 6y - 8z = 20
Equation 3: 4x - y - 8z = 6

Step 2: Eliminate x from equations 1 and 2.
Multiply equation 1 by 2: 6x - 10y + 8z = -24
Multiply equation 2 by 3: -6x + 18y - 24z = 60
Add:
6x - 6x = 0
-10y + 18y = 8y
8z - 24z = -16z
-24 + 60 = 36
So: 8y - 16z = 36 → divide by 4: 2y - 4z = 9 → Equation A

Step 3: Eliminate x from equations 1 and 3.
Multiply equation 1 by 4: 12x - 20y + 16z = -48
Multiply equation 3 by 3: 12x - 3y - 24z = 18
Subtract:
12x - 12x = 0
-20y + 3y = -17y
16z + 24z = 40z? Wait, no: 16z - (-24z)? No, it's 16z - 24z = -8z
Wait, subtracting: (12x - 20y + 16z) - (12x - 3y - 24z) = -48 - 18
12x - 12x = 0
-20y - (-3y) = -17y
16z - (-24z) = 16z + 24z = 40z
-48 - 18 = -66
So: -17y + 40z = -66 → Equation B

Now we have:
A) 2y - 4z = 9
B) -17y + 40z = -66

Step 4: Eliminate y. Multiply A by 17: 34y - 68z = 153
Multiply B by 2: -34y + 80z = -132
Add:
34y - 34y = 0
-68z + 80z = 12z
153 - 132 = 21
So: 12z = 21 → z = 21/12 = 7/4

Step 5: Plug z = 7/4 into A: 2y - 4(7/4) = 9 → 2y - 7 = 9 → 2y = 16 → y = 8

Step 6: Plug y = 8, z = 7/4 into equation 1: 3x - 5(8) + 4(7/4) = -12
3x - 40 + 7 = -12
3x - 33 = -12
3x = 21 → x = 7

Check in equation 2: -2(7) + 6(8) - 8(7/4) = -14 + 48 - 14 = 20 → Correct
Check in equation 3: 4(7) - 8 - 8(7/4) = 28 - 8 - 14 = 6 → Correct

So solution for System 5: x = 7, y = 8, z = 7/4

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System 6:
1) -2x + 2y + z = 14
2) 3x - 2y + z = -5
3) -x + y - 2z = -8

Step 1: Eliminate z from equations 1 and 2.
Subtract equation 1 from equation 2:
(3x - 2y + z) - (-2x + 2y + z) = -5 - 14
3x + 2x = 5x
-2y - 2y = -4y
z - z = 0
-5 - 14 = -19
So: 5x - 4y = -19 → Equation A

Step 2: Eliminate z from equations 1 and 3.
Multiply equation 3 by 1: -x + y - 2z = -8
Multiply equation 1 by 2: -4x + 4y + 2z = 28
Add:
-4x - x = -5x
4y + y = 5y
2z - 2z = 0
28 - 8 = 20
So: -5x + 5y = 20 → divide by 5: -x + y = 4 → Equation B

Now we have:
A) 5x - 4y = -19
B) -x + y = 4

From B: y = x + 4

Plug into A: 5x - 4(x + 4) = -19 → 5x - 4x - 16 = -19 → x - 16 = -19 → x = -3

Then y = -3 + 4 = 1

Now plug x = -3, y = 1 into equation 1: -2(-3) + 2(1) + z = 14 → 6 + 2 + z = 14 → 8 + z = 14 → z = 6

Check in equation 2: 3(-3) - 2(1) + 6 = -9 - 2 + 6 = -5 → Correct
Check in equation 3: -(-3) + 1 - 2(6) = 3 + 1 - 12 = -8 → Correct

So solution for System 6: x = -3, y = 1, z = 6

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Final Answer:
System 1: x = 3, y = 1, z = -2
System 2: x = 0, y = 0, z = -3
System 3: x = 5, y = 4, z = -5
System 4: x = 43/3, y = 68/3, z = -8/3
System 5: x = 7, y = 8, z = 7/4
System 6: x = -3, y = 1, z = 6