Linear Equations

Scalar linear equation with one unknown

The basic form of linear scalar equations with constant coefficients a and b and the variable x is:

$a \cdot x = b$

The solution of the equation is obtained by dividing the equation by the coefficient a.

$x = \frac{b}{a}$

Examples

Example: Linear equation in normal form

$2 \cdot x = 6$

$x = \frac{6}{2} = 3$

The solution of the equation is obtained by dividing the equation by 2.

Example: conversion to normal form

$4 \cdot x - 4 = 2 \cdot x - 5$

$2 \cdot x - 4 = - 5$

$2 \cdot x = - 1$

$x = - \frac{1}{2}$

1 Forming: subtraction 2x

2 Forming: addition +4. Thus, the normal form is reached.

3 Forming: Division by 2 leads to the solution.

Example: conversion of a fraction

$5 + \frac{1}{2 \cdot x} = 2$

$10 \cdot x + 1 = 4 \cdot x$

$6 \cdot x + 1 = 0$

$6 \cdot x = - 1$

$x = - \frac{1}{6}$

1 Forming: multiplication by 2x

3 Forming: subtraction of 1 leads to the normal form.

4 Forming: Division by 6 gives the solution.

Example: The Unknown in fractions

$\frac{2}{x} + \frac{1}{2 \cdot x} = 2$

$\frac{4}{2 \cdot x} + \frac{1}{2 \cdot x} = 2$

$\frac{4 + 1}{2 \cdot x} = 2$

$5 = 4 \cdot x$

$x = \frac{5}{4}$

1 Forming: extension to the common denominator of the first fraction by 2x

2 Forming: fractions on main denominator.

3 Forming: 2x multiplication leads to a normal form.

4 Forming: Division by 4 gives the solution.

Calculator: Linear equation in one variable

$a \cdot x = b$

Scalar linear equation with two unknowns

The basic form of linear scalar equations with constant coefficients a, b and c and the variables x and y is:

$a \cdot x + b \cdot y = c$

For a and b unequal to 0, the equation has a one-dimensional solution space. Solving the equation for y is a linear equation.

$y = \frac{c}{b} - \frac{a}{b} \cdot x$

Substitution with m = a / n and b = c / b results in the line equation with the gradient m and intercept n

$y = n + m \cdot x$

Calculator: Linear equation with two variables

$a \cdot x + b \cdot y = c$

More Calculators

Here is a list of of further useful calculators and sites:

Index Linear Equation Systems Calculator 2x2 systems Calculator 3x3 systems Calculator NxN Cramer's rule Calculator NxN Gauss method Matrix Determinant