Math worksheet for solving systems of equations with three variables.

Let’s solve each system step by step. We’ll use substitution or elimination — whichever is easier for each problem.

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Problem 1:

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

Step 1: Add equations ① and to eliminate x.

+ ③:
(x - x) + (3y + 2y) + (-3z - z) = 12 + 1
→ 5y - 4z = 13 → Call this Equation A

Step 2: Use equation ② and eliminate x with another combo.

Multiply equation ① by 3:
3x + 9y - 9z = 36 → Call this ①×3

Now subtract equation ② from it:

(3x + 9y - 9z) - (3x - y + 4z) = 36 - 0
→ 3x - 3x + 9y + y - 9z - 4z = 36
→ 10y - 13z = 36 → Call this Equation B

Now we have:

A: 5y - 4z = 13
B: 10y - 13z = 36

Multiply Equation A by 2:
10y - 8z = 26 → Call this A×2

Subtract A×2 from B:

(10y - 13z) - (10y - 8z) = 36 - 26
→ -5z = 10 → z = -2

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

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 all equations:

①: 3 + 3(1) - 3(-2) = 3+3+6=12 ✔️
②: 3(3) -1 +4(-2)=9-1-8=0 ✔️
③: -3 +2(1) -(-2)= -3+2+2=1 ✔️

Answer for #1: x=3, y=1, z=-2

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Problem 2:

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

Step 1: Subtract ① from ② to eliminate x.

② - ①:
(x - x) + (y - (-6y)) + (-4z - 4z) = 12 - (-12)
→ 7y - 8z = 24 → Equation A

Step 2: Eliminate x using ① and ③.

Multiply ① by 2:
2x - 12y + 8z = -24 → ①×2

Subtract from ③:

(2x + 2y + 5z) - (2x - 12y + 8z) = -15 - (-24)
→ 0x + 14y - 3z = 9 → Equation B

Now:

A: 7y - 8z = 24
B: 14y - 3z = 9

Multiply A by 2:
14y - 16z = 48 → A×2

Subtract B from A×2:

(14y - 16z) - (14y - 3z) = 48 - 9
→ -13z = 39 → z = -3

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

Plug y=0, z=-3 into ②:
x + 0 -4(-3) = 12 → x + 12 = 12 → x = 0

✔ Check:

①: 0 -0 +4(-3)= -12 ✔️
②: 0 +0 -4(-3)=12 ✔️
③: 0 +0 +5(-3)= -15 ✔️

Answer for #2: x=0, y=0, z=-3

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Problem 3:

Equations:
① -2x - y - 4z = 6
② -4x - 2z = -10 → simplify: divide by -2 → 2x + z = 5 → Equation C
③ 5x + y + 6z = -1

Step 1: Add ① and ③ to eliminate y.

① + ③:
(-2x + 5x) + (-y + y) + (-4z + 6z) = 6 + (-1)
→ 3x + 2z = 5 → Equation D

Now we have:

C: 2x + z = 5
D: 3x + 2z = 5

Solve C for z: z = 5 - 2x

Plug into D:
3x + 2(5 - 2x) = 5 → 3x + 10 - 4x = 5 → -x + 10 = 5 → -x = -5 → x = 5

Then z = 5 - 2(5) = 5 - 10 = -5

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

✔ Check:

①: -10 -4 -4(-5)? Wait — let's recalculate:

Wait! Equation ①: -2x - y -4z = 6
x=5, y=4, z=-5 → -2(5) = -10; -y = -4; -4z = -4*(-5)=+20 → total: -10 -4 +20 = 6 ✔️

②: -4(5) -2(-5) = -20 +10 = -10 ✔️
③: 5(5) +4 +6(-5)=25+4-30= -1 ✔️

Answer for #3: x=5, y=4, z=-5

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Problem 4:

Equations:
① -2x + 2y + z = 14
② 3x - 2y + z = -5
③ -x + y + 2z = 3

Step 1: Add ① and ② to eliminate y.

① + ②:
(-2x + 3x) + (2y - 2y) + (z + z) = 14 + (-5)
→ x + 2z = 9 → Equation A

Step 2: Use ③ and eliminate y with another pair.

Multiply ③ by 2:
-2x + 2y + 4z = 6 → ③×2

Subtract ① from this:

(③×2) - ①:
[-2x + 2y + 4z] - [-2x + 2y + z] = 6 - 14
→ 0x + 0y + 3z = -8 → z = -8/3? That seems messy. Let me check again.

Wait — maybe better to do different combination.

Try adding ① and ② gave us x + 2z = 9 → good.

Now take ③: -x + y + 2z = 3 → solve for y: y = x - 2z + 3

Plug into ①:
-2x + 2(x - 2z + 3) + z = 14
→ -2x + 2x -4z + 6 + z = 14
→ -3z + 6 = 14 → -3z = 8 → z = -8/3 → still fractional.

But let’s keep going.

From earlier: x + 2z = 9 → x = 9 - 2z

If z = -8/3, then x = 9 - 2(-8/3) = 9 + 16/3 = 27/3 + 16/3 = 43/3

Then y = x - 2z + 3 = 43/3 - 2(-8/3) + 3 = 43/3 + 16/3 + 9/3 = 68/3

Check in ②: 3x -2y + z = ?

3*(43/3) = 43
-2*(68/3) = -136/3
z = -8/3
Total: 43 - 136/3 - 8/3 = 43 - 144/3 = 43 - 48 = -5 ✔️

So fractions are correct.

But let me double-check if I made a mistake earlier.

Alternative approach:

Add ① and ②: got x + 2z = 9 → OK

Now multiply ③ by 2: -2x + 2y + 4z = 6

Subtract ①: (-2x + 2y + 4z) - (-2x + 2y + z) = 6 - 14 → 3z = -8 → same as before.

So yes, answer is fractional.

But wait — problem 6 is identical to problem 4? Let me check original image.

Looking back at user input:

Problem 4:
-2x + 2y + z = 14
3x - 2y + z = -5
-x + y + 2z = 3

Problem 6:
-2x + 2y + z = 14
3x - 2y + z = -5
-x + y - 2z = -8 ← different last equation!

Ah! So problem 4 and 6 are similar but not same. In problem 4, third equation is “-x + y + 2z = 3”, in problem 6 it’s “-x + y - 2z = -8”

So for problem 4, our solution is:

x = 43/3, y = 68/3, z = -8/3

But that seems odd for a worksheet — maybe I made error?

Wait — let me try solving differently.

Use elimination between ① and ② first.

①: -2x + 2y + z = 14
②: 3x - 2y + z = -5
Add them: x + 2z = 9 → same as before.

Now take ③: -x + y + 2z = 3 → write as y = x - 2z + 3

Plug into ①:
-2x + 2(x - 2z + 3) + z = 14
→ -2x + 2x -4z + 6 + z = 14
→ -3z + 6 = 14 → -3z = 8 → z = -8/3 → same.

Perhaps the problem allows fractions. But let’s hold on — maybe I misread the problem.

Wait — looking back at user’s text:

In problem 4:
“-2x + 2y + z = 14
3x - 2y + z = -5
-x + y + 2z = 3”

Yes.

But let’s verify with these values in ③:

-x + y + 2z = -(43/3) + 68/3 + 2*(-8/3) = (-43 + 68 -16)/3 = 9/3 = 3 ✔️

So it’s correct.

But perhaps the worksheet expects integer answers — maybe I should recheck problem numbering.

Wait — problem 6 has different third equation: “-x + y - 2z = -8”

So for now, proceed.

Answer for #4: x = 43/3, y = 68/3, z = -8/3

But let’s move to problem 5 and 6, then come back if needed.

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Problem 5:

Equations given as:

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

So:

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

Step 1: Notice ② and ③ both have -8z. Subtract them to eliminate z.

③ - ②:
(4x - (-2x)) + (-y - 6y) + (-8z - (-8z)) = 6 - 20
→ 6x -7y + 0z = -14 → Equation A: 6x -7y = -14

Step 2: Make coefficients of z match in ① and .

Multiply ① by 2: 6x -10y +8z = -24 → ①×2

Add to ②:
(6x -10y +8z) + (-2x +6y -8z) = -24 + 20
→ 4x -4y = -4 → divide by 4: x - y = -1 → Equation B

Now we have:

A: 6x -7y = -14
B: x - y = -1 → so x = y -1

Plug into A:
6(y -1) -7y = -14 → 6y -6 -7y = -14 → -y -6 = -14 → -y = -8 → y = 8

Then x = 8 -1 = 7

Plug x=7, y=8 into ①:
3(7) -5(8) +4z = -12 → 21 -40 +4z = -12 → -19 +4z = -12 → 4z = 7 → z = 7/4

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

Answer for #5: x=7, y=8, z=7/4

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Problem 6:

Equations:
① -2x + 2y + z = 14
② 3x - 2y + z = -5
③ -x + y - 2z = -8

This is almost same as problem 4, but third equation changed.

Step 1: Add ① and to eliminate y.

① + ②:
(-2x + 3x) + (2y - 2y) + (z + z) = 14 + (-5)
→ x + 2z = 9 → Equation A

Step 2: Use ③: -x + y - 2z = -8 → solve for y: y = x + 2z - 8

Plug into ①:
-2x + 2(x + 2z - 8) + z = 14
→ -2x + 2x + 4z - 16 + z = 14
→ 5z - 16 = 14 → 5z = 30 → z = 6

Then from A: x + 2(6) = 9 → x + 12 = 9 → x = -3

Then y = x + 2z - 8 = -3 + 12 - 8 = 1

✔ Check:

①: -2(-3) + 2(1) + 6 = 6 + 2 + 6 = 14 ✔️
②: 3(-3) -2(1) +6 = -9 -2 +6 = -5 ✔️
③: -(-3) +1 -2(6) = 3 +1 -12 = -8 ✔️

Perfect integers!

Answer for #6: x=-3, y=1, z=6

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Now go back to Problem 4 — since Problem 6 worked out nicely, maybe I misread Problem 4?

User wrote for Problem 4:

“-2x + 2y + z = 14
3x - 2y + z = -5
-x + y + 2z = 3”

And we got fractions. But let’s see if there’s an error.

Alternative method for Problem 4:

From ① and ② added: x + 2z = 9 → x = 9 - 2z

From ③: -x + y + 2z = 3 → plug x: -(9 - 2z) + y + 2z = 3 → -9 + 2z + y + 2z = 3 → y + 4z = 12 → y = 12 - 4z

Now plug into ①: -2x + 2y + z = 14

x = 9 - 2z, y = 12 - 4z

So: -2(9 - 2z) + 2(12 - 4z) + z = 14
→ -18 + 4z + 24 - 8z + z = 14
→ ( -18 + 24 ) + (4z -8z + z) = 14
→ 6 - 3z = 14 → -3z = 8 → z = -8/3 → same as before.

So it’s correct. Perhaps the worksheet has one fractional answer.

But let me check if the third equation was meant to be “-x + y - 2z = something” — but according to user input, it’s “+2z = 3”.

So we’ll keep it.

Final Answers:

1. x=3, y=1, z=-2
2. x=0, y=0, z=-3
3. x=5, y=4, z=-5
4. x=43/3, y=68/3, z=-8/3
5. x=7, y=8, z=7/4
6. x=-3, y=1, z=6

But for problem 4 and 5, fractions are acceptable unless specified otherwise.

However, let me double-check problem 4 once more with decimal approximations to ensure no arithmetic error.

z = -8/3 ≈ -2.6667
x = 9 - 2*(-8/3) = 9 + 16/3 = 43/3 ≈ 14.333
y = 12 - 4*(-8/3) = 12 + 32/3 = 68/3 ≈ 22.6667

Plug into ①: -2*(43/3) + 2*(68/3) + (-8/3) = (-86 + 136 -8)/3 = 42/3 = 14 ✔️

All good.

So final answers are as above.

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