Linear Equation Calculator
Solve ax + b = cx + d for x with classification (unique, infinite, none) and verification. Free linear equation solver.
Calculator
Both Sides of the Equation
Solution Summary
Results
Solution x
3
Value satisfying both sides
One solution
Classification from k = a − c
2
k = a − c
6
m = d − b
0
Must be 0 at the solution
The solution is verified by evaluating both sides at x; equality within 1e-9 confirms correctness.
Visualizations
Left Side vs Right Side at Solution
Solution Status
Formula
Subtracting cx gathers variables left; subtracting b gathers constants right. One division then isolates x — unless k = 0.
Alternative Form
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| a, b | Left side | numbers | Coefficient and constant of ax + b |
| c, d | Right side | numbers | Coefficient and constant of cx + d |
| k | Net x-coefficient | number | a − c; zero means no unique solution |
| m | Net constant | number | d − b |
Explanation
k measures how much faster the left side grows than the right; m is the head start to cancel. Their ratio is the meeting point.
Derivation
ax + b = cx + d → (a − c)x = d − b → kx = m → x = m/k for k ≠ 0. If k = 0: m = 0 gives identity, m ≠ 0 contradiction.
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Write the equation with the given numbers.
Subtract cx from both sides to collect x on the left.
Subtract b from both sides to isolate the x term.
Divide by k (or classify when k = 0).
Verify by evaluating both sides at x.
Worked Examples
See how the calculator works with different scenarios.
Scenario
x on both sides, small integers.
Inputs
Calculation
2x = 6 → x = 3; check 11 = 11
Result
x = 3
Interpretation
Collecting gives 2x = 6; substitution confirms both sides equal 11.
Understanding Your Result
Your Result: ax + b = cx + d solved with classification and two-sided check
Typical range: Unique solution whenever a ≠ c. Equal slopes give identity (same intercept) or contradiction.
What This Means
x is the point where both linear expressions agree; the case label tells you whether that point is unique, everywhere, or nowhere.
Why It Matters
Break-even, meeting-time and equilibrium problems all equate two linear models — the solution is where they cross.
Professional Interpretation
Always compute k = a − c first: its sign and zero-status predict everything before any division.
Recommendations
- •Compute k = a − c before dividing
- •Verify by evaluating both sides
- •Interpret k = 0 cases honestly
- •Track signs when moving terms
- •State break-even meaning in word problems
Educational Content
Introduction
A linear equation in one variable equates two first-degree expressions; solving finds every x making them equal.
Concept Explanation
Equality is preserved by doing the same operation to both sides. Collect variables left, constants right, divide once.
Applications
Benefits
- •Automatic case classification
- •Two-sided verification
- •Steps mirror the balance method
- •Collected form shown explicitly
- •Handles decimals directly
Limitations
- •One variable only
- •Linear (degree 1) sides
- •Numeric coefficients
Best Practices
- •Move variables left, constants right
- •Combine before dividing
- •Never divide by zero k
- •Check by substitution
- •Interpret the case, not just the number
Common Misconceptions
Practical Tips
- •Collect x-terms left first
- •Combine fully before dividing
- •Read k = a − c as the net slope
- •Verify both sides numerically
- •Name the k = 0 case correctly
- •Mind sign flips on transposition
- •Attach units in story problems
Common Mistakes
Assumptions
This calculator makes the following assumptions to provide estimates:
Limitations
Be aware of these limitations when using this calculator:
Frequently Asked Questions
Answers to common questions about this calculator.
How do I solve ax + b = cx + d?
What if x appears on both sides?
When does a linear equation have no solution?
When are there infinitely many solutions?
How do I check my answer?
Why do moved terms change sign?
Can coefficients be decimals?
What does the solution mean graphically?
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Related Guides
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Glossary
Technical terms used in mortgage calculations explained in plain language.
- Linear equation
- Equality of two first-degree expressions in one variable.
- Collect like terms
- Gathering x-terms and constants per side.
- Transposition
- Moving a term across = with flipped sign.
- Identity
- Equation true for every x (0 = 0).
- Contradiction
- Equation true for no x (0 = 4).
- Net coefficient
- k = a − c after collecting.
- Back-substitution
- Plugging the solution into both sides.
- Break-even point
- Where two linear models agree.
- Unique solution
- Exactly one x satisfies the equation.
Authoritative References
Trusted sources and standards referenced by this calculator.
Applicable Standards
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Page Information
2026-09-29
2.0.0
Free to use for educational and professional purposes.