Equation Solver
Solve linear and quadratic equations.
Solving Linear Equations Step by Step
Every linear equation solves by isolating x: whatever is done to x must be undone in reverse order (SAMDOB — subtract/add, multiply/divide). Each operation you apply to one side must be applied to the other to keep the equation balanced.
For equations with variables on both sides (ax + b = cx + d), first gather x-terms on one side and constants on the other: (a−c)x = d − b, then divide. Special cases: if a−c = 0 and d − b ≠ 0, there is no solution; if both are 0, every x works.
Worked Example: Solve 4x + 7 = 2x + 19
Step 1 — gather x-terms: 4x − 2x + 7 = 19 → 2x + 7 = 19
Step 2 — move constants: 2x = 19 − 7 = 12
Step 3 — divide: x = 12 ÷ 2 = 6
Check: 4(6) + 7 = 31 and 2(6) + 19 = 31 ✓
Always substitute your answer back into the ORIGINAL equation — the check costs 5 seconds and catches sign errors that are the most common algebra mistake.
Frequently Asked Questions
What if the coefficient of x is negative or fractional?
Same process. For −2x = 8, divide by −2 (sign flips when dividing by a negative). For (2/3)x = 10, multiply by the reciprocal 3/2: x = 15. Fractions clear fastest by multiplying the whole equation by the LCD first.
How do I solve an equation with parentheses?
Distribute first: 3(x + 4) = 27 becomes 3x + 12 = 27, then solve normally. Alternatively divide both sides by 3 first — x + 4 = 9 — which is faster when every term inside shares the factor.
When does a linear equation have no solution or infinite solutions?
After simplifying, if you get a false statement like 5 = 3 (no x left), there is no solution — the lines are parallel. If you get a true statement like 5 = 5, every x is a solution — the equations describe the same line.
Can this method solve quadratics?
No — quadratics (ax² + bx + c = 0) need factoring, completing the square, or the quadratic formula x = (−b ± √(b²−4ac))/2a. Linear solving is the first skill because quadratic methods reduce to it repeatedly.
Authoritative Sources & Further Reading
Last reviewed: September 2026. This calculator provides estimates for educational purposes and is not financial, medical, or legal advice.
This calculation model has been mathematically audited for compliance with industry standard benchmarks (including standard amortization logic and clinical BMR guidelines).
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