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Three-Variable Systems in Matrix Form

Ages 14–15

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Three-Variable Systems in Matrix Form

Three equations in x, y and z can be written AX = B, where A is a 3 × 3 coefficient matrix, X is [[x], [y], [z]], and B is a 3 × 1 constants matrix. Check the determinant or use row reduction to decide whether a unique solution exists. Eliminate one variable at a time, solve the resulting two-variable system, then substitute back for the final variable.

Your goals

  • ✓You can set up and solve three simultaneous equations using matrices.
  • ✓You can use coefficient matrix, free variable, consistent correctly.
  • ✓You can keep x, y, z in the same column order and substitute into all three equations.

Words to know

coefficient matrix:
The 3 × 3 matrix of x, y, and z coefficients.
free variable:
A variable whose column has no pivot and can be chosen as a parameter.
consistent:
A system with at least one solution.

Examples

  • 1x + y + z = 6, 2x − y + z = 3 and x + 2y − z = 2 become 1112−1112−1xyz = 632.
  • 2The coefficient order must be x, y, z in every row; do not place a z coefficient in the y column.
  • 3If row reduction produces pivots in all three variable columns, the system has a unique solution.
  • 4If a variable column has no pivot, introduce a parameter and describe the family of solutions rather than inventing a single answer.
  • 5Substitute the final x, y and z into all three original equations. One correct equation is not enough to verify a system.

More than one way to solve it

Try each method, then choose the one you can explain and check most clearly.

Method 1

Keep variable columns fixed

Best for: Building a 3 × 3 coefficient matrix
  1. 1.Choose one order, such as x, y, z.
  2. 2.Place every equation's coefficients in that order.
  3. 3.Put constants in a separate 3 × 1 column.

Example: A row [2, −1, 3] means 2x − y + 3z in the chosen order.

Method 2

Eliminate then substitute

Best for: Solving three unknowns
  1. 1.Eliminate one variable from two rows.
  2. 2.Solve the resulting two-variable pair.
  3. 3.Substitute both values into an original equation.

Example: Solve x and y first, then use one original row to find z.

Check these

  • • Do not reorder variables in one row or claim a unique solution without three pivots.
  • • Keep every number, symbol, label, and unit in its correct place.
  • • Choose the method that makes each step easiest to explain and check.

Helpful habits

  • • Read the whole question before choosing what to do.
  • • Show one clear step at a time and check each step.
  • • If you are stuck, draw a model, try smaller numbers, or revisit the example.

Now try your own

Create your own three-variable systems in matrix form example, solve it, and explain how you checked it.

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