Systems of Linear Equations Worksheets with Answer Key - Free Printable

Since the user has set all options (ExplainAnswer, TeachTopic, GeneratePractice) to false, I will proceed directly to solving one of the systems from the image as requested by the core instruction: “Solve the problem accurately.”

I’ll choose Problem #1 for demonstration:

Problem #1:
3x + 3y - 3z = 21
5x - 4y - 2z = 2
4x + 2y - 3z = 21

Step 1: Simplify Equation 1 if possible.
Equation 1: 3x + 3y - 3z = 21 → divide entire equation by 3 → x + y - z = 7
Let’s call this Eq1_simplified: x + y - z = 7

Now our system is:
Eq1_simp: x + y - z = 7
Eq2: 5x - 4y - 2z = 2
Eq3: 4x + 2y - 3z = 21

Step 2: Eliminate one variable using two equations. Let’s eliminate ‘z’ first.

Use Eq1_simp and Eq2 to eliminate z.
From Eq1_simp: x + y - z = 7 → multiply by 2 → 2x + 2y - 2z = 14 → call this EqA
Eq2: 5x - 4y - 2z = 2 → keep as is → EqB

Subtract EqA from EqB:
(5x - 4y - 2z) - (2x + 2y - 2z) = 2 - 14
→ 5x - 4y - 2z - 2x - 2y + 2z = -12
→ 3x - 6y = -12
Divide by 3: x - 2y = -4 → call this EqC

Step 3: Use Eq1_simp and Eq3 to eliminate z again.
Eq1_simp: x + y - z = 7 → multiply by 3 → 3x + 3y - 3z = 21 → call this EqD
Eq3: 4x + 2y - 3z = 21 → keep as is → EqE

Subtract EqD from EqE:
(4x + 2y - 3z) - (3x + 3y - 3z) = 21 - 21
→ 4x + 2y - 3z - 3x - 3y + 3z = 0
→ x - y = 0 → so x = y → call this EqF

Step 4: Substitute EqF into EqC.
EqC: x - 2y = -4
But x = y, so substitute: y - 2y = -4 → -y = -4 → y = 4
Then since x = y → x = 4

Step 5: Plug x=4, y=4 into Eq1_simp to find z.
Eq1_simp: x + y - z = 7 → 4 + 4 - z = 7 → 8 - z = 7 → z = 1

Step 6: Verify solution in original equations.

Check Eq1: 3x + 3y - 3z = 3(4)+3(4)-3(1)=12+12-3=21 ✔
Check Eq2: 5x -4y -2z = 5(4)-4(4)-2(1)=20-16-2=2 ✔
Check Eq3: 4x +2y -3z = 4(4)+2(4)-3(1)=16+8-3=21 ✔

All equations satisfied.

Final Answer:
x = 4, y = 4, z = 1