Linear Equation Calculator
What is an Equation?
In mathematics, an equation is a statement that asserts the equality of two expressions, which are connected by the equals sign "=". The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any equality is an equation.
Solving an equation containing variables consists of determining which values of the variables make the equality true. Variables 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 variable. A conditional equation is only true for particular values of the variables.
An equation is written as two expressions, connected by an equals sign ("="). The expressions on the two sides of the equals sign are called the "left-hand side" and "right-hand side" of the equation.
In mathematics, a linear equation is an equation that may be put in the form:
3x+ 2 = 14
Where x is the variable (or unknown), and 3 is the coefficient, which are often real numbers. The coefficients may be considered as parameters of the equation, and may be arbitrary expressions, provided they do not contain any of the variables.
Alternatively a linear equation can be obtained by equating to zero a linear polynomial over some field, from which the coefficients are taken.
The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true.
In the case of just one variable, the term linear equation refers implicitly to this particular case, in which the variable is sensibly called the unknown.
In the case of two variables, each solution may be interpreted as the Cartesian coordinates of a point of the Euclidean plane. The solutions of a linear equation form a line in the Euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables. This is the origin of the term linear for describing this type of equations. More generally, the solutions of a linear equation in n variables form a hyperplane (a subspace of dimension n − 1) in the Euclidean space of dimension n.
Linear equations occur frequently in all mathematics and their applications in physics and engineering, partly because non-linear systems are often well approximated by linear equations.
This article considers the case of a single equation with coefficients from the field of real numbers, for which one studies the real solutions. All of its content applies to complex solutions and, more generally, for linear equations with coefficients and solutions in any field. For the case of several simultaneous linear equations, see system of linear equations.
Frequently the term linear equation refers implicitly to the case of just one variable.
In this case, the equation can be put in the form:
ax + b = 0
and it has a unique solution:
x= - b/a
in the general case where a ≠ 0. In this case, the name unknown is sensibly given to the variable x.
If a = 0, there are two cases. Either b equals also 0, and every number is a solution. Otherwise b ≠ 0, and there is no solution. In this latter case, the equation is said to be inconsistent.
In the case of two variables, any linear equation can be put in the form:
ax + by + c = 0
where the variables are x and y, and the coefficients are a, b and c.
An equivalent equation (that is an equation with exactly the same solutions) is:
Ax + By = C
with A = a, B = b, and C = –
These equivalent variants are sometimes given generic names, such as general form or standard form.
There are other forms for a linear equation (see below), which can all be transformed in the standard form with simple algebraic manipulations, such as adding the same quantity to both members of the equation, or multiplying both members by the same nonzero constant.
If b ≠ 0, the equation,
ax + by + c = 0
is a linear equation in the single variable y for every value of x. It has therefore a unique solution for y, which is given by:
y= - ax/b - c/b
This defines a function.
The functions whose graph is a line are generally called linear functions in the context of calculus.
However, in linear algebra, a linear function is a function that maps a sum to the sum of the images of the summands. So, for this definition, the above function is linear only when c = 0, that is when the line passes through the origin. For avoiding confusion, the functions whose graph is an arbitrary line are often called affine functions.
Each solution (x, y) of a linear equation:
ax + by + c = 0
ax+by+c=0 may be viewed as the Cartesian coordinates of a point in the Euclidean plane. With this interpretation, all solutions of the equation form a line, provided that a and b are not both zero. Conversely, every line is the set of all solutions of a linear equation.
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. If b = 0, the line is a vertical line (that is a line parallel:
x= - c/a
to the y-axis) of equation x=−ca, which is not the graph of a function of x.
Similarly, if a ≠ 0, the line is the graph of a function of y, and, if a = 0, one has a horizontal line of equation:
y= - c/b