Random Linear Equation Exercise Generator

Generate nonsingular two-variable linear systems from integer solutions and compute their answer keys.

Description

Construct uniquely solvable two-variable linear systems from bounded integer solutions and return answer keys.

A system of two linear equations asks for an ordered pair satisfying both equations. A nonzero coefficient determinant ensures a unique solution.^[1]

This generator chooses integer x and y first, then constructs two equations with coefficient rows [a, b] and [−b, a]. Each result.exercises item has input containing a1, b1, c1, a2, b2, c2, and solution containing x, y, and determinant.

When to use Random Linear Equation Exercise Generator

Use it for elimination, substitution, or determinant-based solution worksheets with a supplied answer key.

How to use Random Linear Equation Exercise Generator and understand its results

Enter integer exerciseCount from 1 through 64 and solutionMagnitude from 1 through 1,000.
Run and interpret input as a1 × x + b1 × y = c1 and a2 × x + b2 × y = c2.
Keep solution separate from the learner-facing equations, then check a response by substitution or compare the returned x and y.

x,y = −M + floor(randomDraw × (2M + 1)); a ∈ {1,…,9}; b ∈ {−9,…,9}

Four draws independently choose the two solution coordinates and the two coefficients.

a1 = a; b1 = b; c1 = ax + by; a2 = −b; b2 = a; c2 = −bx + ay

The coefficient determinant is a² + b², strictly positive because a cannot be zero.

The linear-equation calculator applies Cramer’s rule to the constructed coefficients and returns the solution and determinant. The coefficient rows are perpendicular and have equal length; this is a constrained family of systems.

json

{
  "input": {
    "exerciseCount": 3,
    "solutionMagnitude": 10
  },
  "randomDraw": 0.25,
  "output": {
    "result": {
      "exercises": [
        {
          "input": {
            "a1": 3,
            "b1": -5,
            "c1": 10,
            "a2": 5,
            "b2": 3,
            "c2": -40
          },
          "solution": {
            "determinant": 34,
            "x": -5,
            "y": -5
          }
        },
        {
          "input": {
            "a1": 3,
            "b1": -5,
            "c1": 10,
            "a2": 5,
            "b2": 3,
            "c2": -40
          },
          "solution": {
            "determinant": 34,
            "x": -5,
            "y": -5
          }
        },
        {
          "input": {
            "a1": 3,
            "b1": -5,
            "c1": 10,
            "a2": 5,
            "b2": 3,
            "c2": -40
          },
          "solution": {
            "determinant": 34,
            "x": -5,
            "y": -5
          }
        }
      ]
    }
  }
}

Zero solution coordinates and b = 0 are permitted. The magnitude bounds x and y, not the right-hand sides, which can exceed it. Singular, inconsistent, and dependent systems cannot be produced by this construction.

Limitations and assumptions

The default source is Math.random; no seed input or cryptographic entropy guarantee is provided. Repeated runs may select the same result, and the distribution depends on the random source.

There are no fractional or complex solutions, arbitrary independently sampled coefficient rows, difficulty grading, diagrams, or automatically hidden answers. Repeated exercises are allowed.

Alternative or Complementary methods

Use the deterministic solver for your own coefficients. A quadratic generator creates a different equation family with bounded integer roots.

References

College Algebra with Corequisite Support 2e: Systems of Linear Equations, Two Variables — OpenStax, Rice University

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