Enter equation values
This calculator solves equations in the form (ax + b) / (cx + d) = (ex + f) / (gx + h).
Formula used
For a fraction equation in the form (ax + b) / (cx + d) = (ex + f) / (gx + h), first apply domain restrictions where cx + d ≠ 0 and gx + h ≠ 0.
Then cross multiply:
After expansion and rearrangement, the calculator builds:
Where:
- A = ag − ec
- B = ah + bg − ed − fc
- C = bh − fd
If A = 0, the equation is linear and x = −C / B. If A ≠ 0, the roots are found using the quadratic formula:
How to use this calculator
- Enter the four coefficients for the left fraction: a, b, c, and d.
- Enter the four coefficients for the right fraction: e, f, g, and h.
- Choose how many decimal places you want in the output.
- Click Solve Equation to generate the result above the form.
- Review restrictions, transformed equation, solutions, and verification values.
- Use the CSV or PDF button to save the result for practice or reporting.
Example data table
| Example | Left fraction | Right fraction | Restrictions | Result summary |
|---|---|---|---|---|
| Example 1 | (2x + 3) / (x - 4) | (x + 5) / 2 | x ≠ 4 | x = -1 and x = 8 |
| Example 2 | (x + 1) / (x - 2) | (x + 1) / (x - 2) | x ≠ 2 | Identity, all allowed x values |
| Example 3 | (x + 2) / (x - 1) | (x + 2) / (x - 1) + impossible match | x ≠ 1 | No valid solution when transformed constant stays nonzero |
Frequently asked questions
1. What type of equation does this solver handle?
This tool solves rational equations built from two linear fractions set equal to each other. Each numerator and denominator is entered as a linear expression in x.
2. Why does the calculator show restricted values?
Restricted values make a denominator equal zero. Those x values are excluded before accepting any root, because the original fraction would be undefined there.
3. What is an extraneous solution?
An extraneous solution appears after algebraic manipulation but fails the original equation or violates a denominator restriction. The solver flags it in the verification table.
4. Why can the result become a quadratic equation?
Cross multiplying two linear fractions creates products of linear expressions. Expanding them can produce an x² term, which turns the cleared equation into a quadratic.
5. Can the calculator return no real solution?
Yes. A negative discriminant means the cleared equation has no real roots. The solver reports that result while still showing restrictions and algebraic steps.
6. What happens when both sides are identical?
If the transformed coefficients all reduce to zero, the equation is an identity. That means every allowed x value is a solution except restricted values.
7. Can I use decimals instead of integers?
Yes. The inputs accept decimal values, and the calculator formats solutions to your chosen precision. This is useful for modeling applied or classroom examples.
8. What do the CSV and PDF options save?
The exports save the entered equation, equation type, restrictions, coefficients, solutions, rejected roots, and the step list for review or sharing.