LINEAR SYSTEM
Systems of two equations, eliminate and solve.
Fill the four coefficients a, b, c, d and the constants e, f of ax + by = e and cx + dy = f; we eliminate to get x and y. All three cases are detected: unique solution (non-zero determinant), no solution (parallel lines) and infinitely many (coincident lines). Fractional answers are reduced, and the working shows both elimination and Cramer's rule.
① ax + by = e
② cx + dy = f
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How to use
Fill the six numbers for ax + by = e and cx + dy = f (each row: x coefficient, y coefficient, constant) and solve. If the determinant ad − bc = 0 there is no unique solution: parallel-but-different lines give "no solution", identical lines give "infinitely many". The working shows elimination (the school method) and Cramer's rule, and fractions are reduced. Practice generates random systems with integer solutions.
常见问题
When is there no unique solution?
When the determinant D = ad − bc = 0 the lines are parallel or coincident. If the constants also line up the lines coincide and there are infinitely many solutions; otherwise they are parallel and never meet, so there is no solution. There is no fourth case: a linear system has exactly one, none, or infinitely many solutions.
Should I use elimination or Cramer's rule?
Elimination (adding or subtracting equations) is the school method you must be able to use in exams, and it is the first step of the working here. Cramer's rule (dividing by the determinant) is faster with messy coefficients. Both are shown so you can cross-check.
Why are the answers often fractions?
Integer coefficients do not guarantee integer solutions. x + 2y = 5 and 2x + 3y = 9 give x = 3, y = 1, while 2x + 3y = 7 and 3x + 5y = 12 give x = −1, y = 3. Fractions are perfectly normal; this tool reduces them to lowest terms.
Is any data uploaded?
No. All solving runs locally in the browser.