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

$a\xb7x=b$

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

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

Example: Linear equation in normal form

$2\xb7x=6$

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

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

Example: conversion to normal form

$4\xb7x-4=2\xb7x-5$

$2\xb7x-4=-5$

$2\xb7x=-1$

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

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

$5+\frac{1}{2\xb7x}=2$

$10\xb7x+1=4\xb7x$

$6\xb7x+1=0$

$6\xb7x=-1$

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

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

$\frac{2}{x}+\frac{1}{2\xb7x}=2$

$\frac{4}{2\xb7x}+\frac{1}{2\xb7x}=2$

$\frac{4+1}{2\xb7x}=2$

$5=4\xb7x$

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

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.

$a\xb7x=b$

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

$a\xb7x+b\xb7y=c$

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

$y=\frac{c}{b}-\frac{a}{b}\xb7x$

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

$y=n+m\xb7x$

$a\xb7x+b\xb7y=c$

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