Equation Solver: How to Solve Linear, Quadratic, and Systems of Equations Step by Step

Equations appear everywhere — calculating loan payments, analyzing circuits, solving physics problems, or optimizing a business budget. Whether you are a student working through algebra or a professional needing quick answers, knowing how to solve different types of equations efficiently saves hours of manual work. Use the Equation Solver on Today Calculator for instant results with complete step-by-step solutions.

Solving Linear Equations (Ax + B = C)

Linear equations are the most common type. The goal is to isolate the variable (x) on one side.

• Step 1: Move constants to the right side. For 3x + 5 = 20, subtract 5 from both sides: 3x = 15.
• Step 2: Divide by the coefficient. 3x ÷ 3 = 15 ÷ 3, so x = 5.
• Verification: Plug x = 5 back in: 3(5) + 5 = 15 + 5 = 20 ✓

Real-world example: You have $200 for a party. Rentals cost $80, and each guest costs $15 for food. How many guests can you invite? 80 + 15x = 200 → 15x = 120 → x = 8 guests.

Solving Quadratic Equations (Ax² + Bx + C = 0)

Quadratics appear in physics (projectile motion), finance (profit maximization), and geometry (area optimization). Three methods:

Method	When to Use	Example: x² − 5x + 6 = 0
Factoring	When factors are obvious	(x − 2)(x − 3) = 0 → x = 2 or x = 3
Quadratic Formula	Always works	x = [5 ± √(25 − 24)] / 2 = (5 ± 1) / 2 → x = 2, 3
Completing Square	When coefficient is 1	(x − 2.5)² − 0.25 = 0 → x = 2 or x = 3

Quadratic formula: x = [−B ± √(B² − 4AC)] / 2A. The discriminant (B² − 4AC) tells you how many solutions: positive = 2 real solutions, zero = 1 solution, negative = no real solutions.

Real-world example: A ball thrown upward at 40 ft/s from 6 ft high follows h = −16t² + 40t + 6. When does it hit the ground? Set h = 0: −16t² + 40t + 6 = 0. Using the formula: t ≈ 2.6 seconds.

Solving Systems of Equations

When you have two equations with two unknowns, use one of these methods:

• Substitution: Solve one equation for one variable, plug into the other. Best when one variable already has coefficient 1.
• Elimination: Add or subtract equations to cancel one variable. Best when coefficients align neatly.
• Matrix method: Use for 3+ variables. Our Algebra Calculator handles this automatically.

Real-world example: A company sells two products. Product A takes 2 hours to make and generates $30 profit. Product B takes 3 hours and generates $50 profit. You have 40 hours and want maximum profit. What is the best mix?

2A + 3B = 40 (time constraint)
We also need a demand constraint — say you can sell at most 15 of each: A ≤ 15, B ≤ 15. The optimal solution is A = 15, B ≈ 3.3, but since you cannot make partial products: A = 14, B = 4 (2×14 + 3×4 = 40). Profit = 14×$30 + 4×$50 = $420 + $200 = $620.

Quick Troubleshooting Guide

Symptom	Likely Cause	Fix
No solution found	Equation is inconsistent (e.g., 0x = 5)	Check if the equation makes sense
Infinite solutions	Both sides are identical (e.g., 0x = 0)	Any value works — the equation is an identity
Negative under square root	No real solutions for quadratic	Look for complex solutions if applicable
Dividing by zero	Variable cancels incorrectly	Check that you did not divide by an expression that could be zero

Need to solve an equation right now? Use the Equation Solver for instant results with step-by-step solutions. For more advanced algebra including matrices and polynomials, try the Algebra Calculator.