# Linear Equations

## Calculator

#### Linear equation in one variable

$a·x=b$

a= b=

#### Linear equation with two variables

$a·x+b·y=c$

a= b= c=

## 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:

$a·x=b$

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

$2·x=6$

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

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

##### Example: conversion to normal form

$4·x-4=2·x-5$

$2·x-4=-5$

1 Forming: subtraction 2x

$2·x=-1$

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

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

3 Forming: Division by 2 leads to the solution.

##### Example: conversion of a fraction

$5+\frac{1}{2·x}=2$

$10·x+1=4·x$

1 Forming: multiplication by 2x

$6·x+1=0$

2 Forming: subtraction 4x.

$6·x=-1$

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

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

4 Forming: Division by 6 gives the solution.

##### Example: The Unknown in fractions

$\frac{2}{x}+\frac{1}{2·x}=2$

$\frac{4}{2·x}+\frac{1}{2·x}=2$

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

$\frac{4+1}{2·x}=2$

2 Forming: fractions on main denominator.

$5=4·x$

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

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

4 Forming: Division by 4 gives the solution.

### 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:

$a·x+b·y=c$

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

$y=\frac{c}{b}-\frac{a}{b}·x$

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

$y=n+m·x$