Because the representation of any linear equation in Cartesian coordinate system is a straight line. Each term that constitutes a linear equation must be a constant or the product of a constant and a variable. And the equation must contain variables, because if there are no variables, only the constant formula is algebraic, not an equation.
Linear equations are also called linear equations, because the representation of any linear equations in Cartesian coordinate system is a straight line. Each term that constitutes a linear equation must be a constant or the product of a constant and a variable. And the equation must contain variables, because if there are no variables, only the constant formula is an arithmetic expression, not an equation.
If a linear equation contains only one variable (x), then the equation is a linear equation with one variable. If it contains two variables (x and y), it is a binary linear equation, and so on.
One-dimensional equation
One-dimensional linear equation refers to an equation with only one variable and at least one linear monomial on both sides of the equal sign. [2]
Formalization of any one-dimensional linear equation
The equation of. Its solution is that
Here's an example:
The solution is:
One-dimensional linear equation is equal to a linear equation: in short, such as
Power greater than or equal to is not allowed.
Note: When a=0
Ax+b=0 is not a linear equation.
if
This equation has infinitely many solutions; If b=0, then this equation has exactly one solution.
Linear equation form
The form is ax+by+...+cz+d=0, and the linear equation about x and y refers to the equation that can be converted into ax+by+c=0 after sorting (where a, b and c are known numbers, and a and b are not 0 at the same time). One-dimensional linear equation is the simplest equation, and its form is AX = B. Because the graph representing linear equation in coordinate system is a straight line, it is called linear equation.
App application
Binary linear simultaneous equation
Binary linear simultaneous equations can be solved by substitution elimination method or addition and subtraction elimination method. [ 1]
Substitution elimination method
The substitution elimination method is to use one of the equations to express an algebraic expression containing one unknown as another unknown. Then substitute into another equation, thus transforming this group of equations into a method of solving two linear equations in one yuan.
For example:
solve
get
Substitute again
that is
So as to discover
Addition, subtraction and elimination
The method of addition, subtraction and elimination is a method of adding or subtracting two equations to eliminate one of the unknowns.
Usually, we first multiply both sides of an equation by a number that is not 0, so that one of the coefficients is the same as that of the other equation. Then add or subtract the two equations.
For example:
Add the two formulas and eliminate x, that is,
So as to discover
get in touch with
Linearization relation
In the example (not a special case), the variable y is a function of x, and the images of the function and the equation are consistent.
Here f has the following characteristics:
f(x+y)=f(x)+f(y)
f(ax)=af(x)
Here a is not a vector.
If a function satisfies this characteristic, it is called a linear function, or more generally, it is called linearization.
Because of the unique nature of linearity, it has superposition effect on solving linear functions in similar equations. This makes the linear equation the easiest to solve and deduce.
Linear equations have important laws in applied mathematics. Using them, it is easy to build models, and in some cases, it can be assumed that the changes of variables are very small, so many nonlinear equations are transformed into linear equations.
Connection with differential
if
, then
So the linear function has no stagnation point, that is, there is no maximum and minimum, and the slope of the linear function is the coefficient of unknown X.