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The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0 , the line is the graph of the function of x that has been defined in the preceding section.
Some equations are true for all values of the involved variables (such as + = +); such equations are called identities. Conditional equations are true for only some values of the involved variables, e.g. x 2 − 1 = 8 {\displaystyle x^{2}-1=8} is true only for x = 3 {\displaystyle x=3} and x = − 3 {\displaystyle x=-3} .
The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for ...
Separation of variables may be possible in some coordinate systems but not others, [2] and which coordinate systems allow for separation depends on the symmetry properties of the equation. [3] Below is an outline of an argument demonstrating the applicability of the method to certain linear equations, although the precise method may differ in ...
[notes 1] A binary form is a form in two variables. A form is also a function defined on a vector space , which may be expressed as a homogeneous function of the coordinates over any basis . A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar.
Regression is a statistical technique used to help investigate how variation in one or more variables predicts or explains variation in another variable. Bivariate regression aims to identify the equation representing the optimal line that defines the relationship between two variables based on a particular data set.
The Euler–Lagrange equation was developed in connection with their studies of the tautochrone problem. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in ...
A model with exactly one explanatory variable is a simple linear regression; a model with two or more explanatory variables is a multiple linear regression. [1] This term is distinct from multivariate linear regression, which predicts multiple correlated dependent variables rather than a single dependent variable. [2]