icon-komplexe-Zahl Math Tutorial: Linear equation

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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 = b a

Examples

Example: Linear equation in normal form

2 · x = 6

x = 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 = - 1 2

3 Forming: Division by 2 leads to the solution.

Example: conversion of a fraction

5 + 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 = - 1 6

4 Forming: Division by 6 gives the solution.

Example: The Unknown in fractions

2 x + 1 2 · x = 2

4 2 · x + 1 2 · x = 2

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

4 + 1 2 · x = 2

2 Forming: fractions on main denominator.

5 = 4 · x

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

x = 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 = c b - a b · x

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

y = n + m · x