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 of the equals sign.

Example: 3x + 7 = 22
Step 1: Subtract 7 from both sides → 3x = 15
Step 2: Divide both sides by 3 → x = 5

Real-world use: A plumber charges a $75 visit fee plus $45 per hour. If your bill is $255, how many hours did they work? 45x + 75 = 255 → 45x = 180 → x = 4 hours.

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

Quadratic equations involve a squared term. The most reliable method is the quadratic formula:

Formula: x = [−B ± √(B² − 4AC)] ÷ 2A

Example: x² + 5x + 6 = 0
A = 1, B = 5, C = 6
Discriminant: B² − 4AC = 25 − 24 = 1
x = [−5 ± √1] ÷ 2 = [−5 ± 1] ÷ 2
x = −2 or x = −3

Real-world use: Calculating the optimal price for maximum revenue. If R(p) = −2p² + 60p, the maximum occurs at the parabola’s vertex: p = −B ÷ 2A = −60 ÷ (−4) = $15 per unit.

Solving Systems of Equations (Two or More Variables)

Method 1: Substitution

Example:
2x + y = 10
x − y = 2

Step 1: From equation 2, y = x − 2
Step 2: Substitute into equation 1: 2x + (x − 2) = 10 → 3x − 2 = 10 → 3x = 12 → x = 4
Step 3: y = 4 − 2 = y = 2
Solution: (4, 2)

Method 2: Elimination

Example:
3x + 2y = 14
x − 2y = 2

Step 1: Add both equations: (3x + 2y) + (x − 2y) = 14 + 2 → 4x = 16 → x = 4
Step 2: Substitute back: 4 − 2y = 2 → −2y = −2 → y = 1
Solution: (4, 1)

Quick Reference: Equation Types

Type	General Form	Solution Method	Example
Linear	Ax + B = C	Isolate x by inverse operations	2x + 3 = 11 → x = 4
Quadratic	Ax² + Bx + C = 0	Quadratic formula / factoring	x² − 5x + 6 = 0 → x = 2, 3
System (2 var)	Two linear equations	Substitution or elimination	See examples above
Exponential	A · B^x = C	Logarithms	2^x = 16 → x = 4

Tips for Avoiding Common Mistakes

Check your signs: −(−5) = +5, not −5. Double negatives cancel out.
Distribute correctly: 2(x + 3) = 2x + 6, not 2x + 3.
Don’t divide by zero: If the equation involves x² = 0, the solution is x = 0, not “no solution.”
Verify your answer: Plug the solution back into the original equation to confirm it works.
Watch for extraneous solutions: Some equations (especially rational or radical ones) produce answers that don’t satisfy the original equation.

For any equation, let the Equation Solver on Today Calculator handle the math — it shows full step-by-step work so you can learn as you go.