This bounded math equation solver online specification handles ax + b = cx + d with four integer coefficient fields. It shows the reduction, classifies the solution set and checks a single solution in the original equation. It is not an unrestricted expression parser or a completed online service.
Enter four coefficients, not an expression string
Use fields a, b, c and d, each an integer from -1000 through 1000. Zero is valid; blank is not zero. The original equation is always ax + b = cx + d.
For 4x + 3 = x + 15, enter a=4, b=3, c=1, d=15. Keep both the entered coefficients and a readable equation visible. Negative coefficients should be displayed clearly rather than hidden by an awkward sign sequence.
This input scope does not accept functions, quadratics, systems or arbitrary typed algebra. Malformed, noninteger or out-of-range entries need a field-specific error and cleared previous outputs.
Reduce the equation without changing its equality
OpenStax explains checking solutions by substitution and distinguishes equations with no solution from those true for every value. We use those concepts for original coefficient examples. Linear Equations in One Variable
Subtract cx and b from both sides:
ax + b = cx + d
(a − c)x = d − b
For the first example:
4x + 3 = x + 15
3x = 12
x = 4
Verify in the original equation: 4×4 + 3 = 19, and 4 + 15 = 19. Both sides agree. Retain the reduction and substitution record, not just the final 4.
Test the zero-coefficient branch first
Let A = a − c and B = d − b. Before any division, inspect A:
| Condition | Supported classification | Reason |
|---|---|---|
| A ≠ 0 | One solution, x = B/A | Division by the nonzero coefficient isolates x. |
| A = 0 and B ≠ 0 | No solution | 0x cannot equal a nonzero constant. |
| A = 0 and B = 0 | Every real x | The reduced equality is true independently of x. |
For 2x + 5 = 2x + 8, cancellation gives 0x = 3, equivalently 5 = 8. No value of x repairs that contradiction.
For 2x + 5 = 2x + 5, cancellation gives 0x = 0. The answer is all real x, not just x = 0. The identical expressions establish the result; testing one value alone would not establish the whole solution set.
These exact comparisons are appropriate for the declared bounded integer coefficients. Do not silently extend the same parser or equality policy to arbitrary floating inputs.
Preserve a fractional answer and check it
Use another original equation, 3x + 1 = x + 6:
2x = 5
x = 5/2 = 2.5
Left: 3×(5/2) + 1 = 17/2 = 8.5
Right: (5/2) + 6 = 17/2 = 8.5
Keep the reduced rational form 5/2 alongside its decimal reference. For other integer inputs, reduce numerator and denominator and keep the sign consistent. A rounded decimal is a display, not a replacement for the exact fraction in the verification trace.
Make the output inspectable
The native worked sheet should hold input coefficients, original equation, A/B reduction, classification, exact solution if unique and substitution evidence. For no-solution and all-real cases, show the contradiction or identity instead of a division error or fabricated single x.
The local HTML inspector keeps those fields together. Invalid input clears the reduction, classification and old substitution result; it does not quietly reinterpret a blank coefficient as 0. Check the local example against the worked cases before using a result.
This packet records no completed homework or learner assessment. It teaches the method for the specified form rather than promising to solve every equation someone pastes online.
Copy this prompt for the equation inspector
Prepare an original US-English specification and native worked sheet for a local linear-equation inspector. Do not implement or claim a tested online solver, expression parser, API or assessment service.
Four input fields a,b,c,d, integers -1000..1000 inclusive, equation ax+b=cx+d. Zero valid; blank/malformed/noninteger/out-of-range invalid, clear all stale reduction/classification/substitution outputs. No arbitrary expression strings, functions, quadratics or systems. Keep coefficients and readable equation/signs visible.
Derive A = a − c and B = d − b, giving Ax = B. Check A = 0 before division: A ≠ 0 means one solution x = B/A; A = 0 and B ≠ 0 means no solution; A = 0 and B = 0 means every real x, not just x = 0. Integer bounds support exact coefficient comparisons. For a unique solution reduce rational numerator/denominator, preserve the exact fraction and label the decimal display; substitute the exact solution into the original sides.
Original audits: 4x + 3 = x + 15 reduces to 3x = 12 and x = 4; both original sides equal 19. 2x + 5 = 2x + 8 reduces to 0x = 3, no solution. 2x + 5 = 2x + 5 reduces to 0x = 0, all real x. 3x + 1 = x + 6 reduces to 2x = 5 and x = 5/2 = 2.5; both original sides equal 17/2 = 8.5. For no/all solution show contradiction/identity, not division or one-value proof. Actual learner records blank. Native sheet authoritative; later input/zero-branch/fraction/keyboard/width tests pending, no persistence or unrestricted capability claim.
Review the classification beside the check
The local HTML example shows the coefficient fields, reduction, classification and substitution trace together. NOS supports saving and rendering prepared HTML. Keep the maintained worked sheet authoritative.
Explore NOS to present the prepared equation review. Leave with a solution set and the equation evidence that supports it, including when there is no single number to report.