Linear Equations

Calculator for solving linear equations with one unknown - the foundation of algebra

Linear Equation Calculator

What is calculated?

This function solves linear equations of the form ax + b = 0 with one unknown x. Linear equations are degree 1 polynomials and form the foundation of algebra.

Coefficients of the linear equation
ax + b = 0
a ≠ 0 for a unique solution
Absolute term
Solution of the linear equation
Unknown x
Linear equations have at most one solution

Linear Equations Info

Properties

Linear equations:

  • Degree 1 polynomial
  • At most one solution
  • Graphically: a line
  • Simple solution formula

Principle: Isolate the unknown using equivalent transformations.

Solution types
Unique solution (a ≠ 0) Infinite solutions (a = b = 0) No solution (a = 0, b ≠ 0)
Examples
2x + 5 = 0:
x = -5/2 = -2.5
3x - 9 = 0:
x = 9/3 = 3
0x + 1 = 0:
No solution
Related topics

→ Quadratic Equations
→ Linear Systems
→ Linear Inequalities


Formulas and theory of linear equations

Standard form
\[ax + b = 0\] Linear equation
Solution formula
\[x = -\frac{b}{a} \quad (a \neq 0)\] Direct solution
General form
\[ax + b = c\] \[x = \frac{c - b}{a}\]
Slope-intercept form
\[y = mx + n\] Line equation
Root
\[f(x) = ax + b\] \[x_0 = -\frac{b}{a}\]
Proportionality
\[y = kx \quad (b = 0)\] Direct proportionality
Point-slope form
\[y - y_1 = m(x - x_1)\] Line through a point
Two-point form
\[y = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) + y_1\] Line through two points

Detailed worked example

Example: 2x + 5 = 0

Given:

  • a = 2 (coefficient of x)
  • b = 5 (constant term)

Equation:

\[2x + 5 = 0\]

Step 1: Isolate x by subtraction

\[2x + 5 = 0\] \[2x = 0 - 5\] \[2x = -5\]

Step 2: Divide by the coefficient

\[x = \frac{-5}{2}\] \[x = -2.5\]

Solution: x = -2.5

Check: 2(-2.5) + 5 = -5 + 5 = 0 ✓

Equivalent transformations

Allowed operations (solution-preserving)
Addition/Subtraction

Add/subtract the same number to both sides

\[ax + b = 0 \quad | \pm c\] \[ax + b \pm c = \pm c\]

Move terms

Multiplication/Division

Multiply/divide both sides by the same number ≠ 0

\[ax + b = 0 \quad | \times k\] \[k(ax + b) = k \cdot 0\]

k ≠ 0 required

Systematic approach
1. Simplify
Expand parentheses
2. Collect
Gather x-terms on one side
3. Isolate
Subtract constants
4. Divide
By coefficient

Geometric interpretation

Linear function as a line
Positive slope

a > 0: Increasing

Negative slope

a < 0: Decreasing

Horizontal line

a = 0: Constant

Root and y-intercept

Root (intersection with x-axis):

\[x_0 = -\frac{b}{a}\]

y-intercept:

\[f(0) = b\]

Practical applications

Economics & Finance
  • Cost functions
  • Profit-loss analysis
  • Break-even point
  • Interest calculations
Physics & Engineering
  • Motion equations
  • Ohm's law
  • Temperature conversions
  • Proportionalities
Everyday & Geometry
  • Scale calculations
  • Conversions
  • Line equations
  • Time calculations

Special cases

Unique solution

Condition: a ≠ 0

\[x = -\frac{b}{a}\]

Exactly one intersection with the x-axis

Infinite solutions

Condition: a = 0 and b = 0

\[0 \cdot x + 0 = 0\]

Every real number is a solution

No solution

Condition: a = 0 and b ≠ 0

\[0 \cdot x + b = 0\]

Contradiction, no solution possible

Historical context

Development of linear algebra
  • Antiquity: Geometric problems and proportions
  • Al-Khwarizmi (~830): Systematic algebra
  • Vieta (16th c.): Symbolic notation
  • Descartes (17th c.): Coordinate geometry
Modern significance
  • Foundation of higher mathematics
  • Linear systems in all sciences
  • Computer graphics and 3D modeling
  • Machine learning and AI
Summary

Linear equations form the foundation of algebra and despite their simplicity are of fundamental importance to all of mathematics. They connect arithmetic operations with geometric concepts and find application in virtually all quantitative sciences.




Mathematical Equations

Associative Property  •  Linear equation  •  Quadratic equation  •  Cubic equation  •  Discriminant  •  Gauss elimination method  •