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Systems of Three Linear Equations

A free College Algebra lesson from the “Systems, Matrices, and Linear Programming” unit, with a worked example and practice problems including step-by-step solutions.

A 3x3 linear system has three variables (typically x, y, z) and three equations. Solve by eliminating one variable from two pairs of equations to reduce to a 2x2 system, then solve that and back-substitute. If the elimination process produces a contradiction (like 0 = 5), the system has no solution. If it produces 0 = 0, there are infinitely many solutions.

What you'll learn

Why it matters: Mixture problems with three components, supply-and-demand with three goods, and traffic-flow at three-way intersections all model with 3x3 linear systems.

Worked example

Problem. Solve x + y + z = 6; 2x - y + z = 3; x + 2y - z = 5. Enter z.

  1. Add eq1 and eq2: 3x + 2z = 9. Add eq1 and eq3: 2x + 3y + 0z = 11 (z cancels).
  2. From eq1: y = 6 - x - z. Substitute and solve. After arithmetic: x = 1, y = 2, z = 3.
  3. Check eq2: 2(1) - 2 + 3 = 3. ✓

Answer: 3

Practice problems

1. If a 3x3 linear system reduces to 0 = 5, how many solutions are there?

Show solution
  1. Warm-up: First identify exactly what the question is asking: If a 3x3 linear system reduces to 0 = 5, how many solutions are there?
  2. Look for a constant rate of change and connect the equation, table, or graph back to that rate.
  3. Contradiction means no solution.
  4. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

2. If a 3x3 system reduces to 0 = 0, how many solutions are there?

Show solution
  1. Warm-up: First identify exactly what the question is asking: If a 3x3 system reduces to 0 = 0, how many solutions are there?
  2. For a system, use substitution, elimination, or graphing to find the value pair that makes both equations true.
  3. A true identity means dependent system.
  4. Check the result by substituting or estimating: the response should match infinitely many and make sense in the original problem.

Answer: infinitely many

3. How many equations are needed for a unique solution in 3 unknowns?

Show solution
  1. Warm-up: First identify exactly what the question is asking: How many equations are needed for a unique solution in 3 unknowns?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Need at least as many independent equations as unknowns.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

4. Solve x + y + z = 6; x - y + z = 2; x + y - z = 0. Enter z.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve x + y + z = 6; x - y + z = 2; x + y - z = 0. Enter z.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Add eq1 + eq3: 2x + 2y = 6 -> x + y = 3.
  4. Subtract eq2 from eq1: 2y = 4 -> y = 2.
  5. Then x = 1 and z = 6 - 1 - 2 = 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

5. Same system. Enter x.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Same system. Enter x.
  2. For a system, use substitution, elimination, or graphing to find the value pair that makes both equations true.
  3. From the work above: x = 1.
  4. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

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