I wanted to know about liner equation?
Answers
In mathematics, a linear equation is an equation that may be put in the form
{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}+b=0,} {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}+b=0,}
where {\displaystyle x_{1},\ldots ,x_{n}} x_{1},\ldots ,x_{n} are the variables (or unknowns or indeterminates), and {\displaystyle b,a_{1},\ldots ,a_{n}} {\displaystyle b,a_{1},\ldots ,a_{n}} are the coefficients, 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. To yield a meaningful equation for non-zero values of {\displaystyle b,} b, the coefficients are required to not being all zero.
In the words of algebra, a linear equation is obtained by equating to zero a linear polynomial over some field, from which the coefficients are taken, and that does not contain the symbols for the indeterminates.
The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true.
The case of just one variable is particularly important, and frequently the term linear equation refers implicitly to this particular case, in which the name unknown for the variable is sensibly used.
All the pairs of numbers that are solutions of a linear equation in two variables form a line in the Euclidean plane, and every non-vertical line may be defined as the solutions of a linear equation. This is the origin of the term linear for describing this type of equation. 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.
A linear equation is an equation for a straight line
These are all linear equations:
yes y = 2x + 1
yes 5x = 6 + 3y
yes y/2 = 3 − x
Let us look more closely at one example:
Example: y = 2x + 1 is a linear equation:
line on a graph
The graph of y = 2x+1 is a straight line
When x increases, y increases twice as fast, so we need 2x
When x is 0, y is already 1. So +1 is also needed
And so: y = 2x + 1
Here are some example values:
x y = 2x + 1
-1 y = 2 × (-1) + 1 = -1
0 y = 2 × 0 + 1 = 1
1 y = 2 × 1 + 1 = 3
2 y = 2 × 2 + 1 = 5
Check for yourself that those points are part of the line above!
Different Forms
There are many ways of writing linear equations, but they usually have constants (like "2" or "c") and must have simple variables (like "x" or "y").
Examples: These are linear equations:
yes y = 3x − 6
yes y − 2 = 3(x + 1)
yes y + 2x − 2 = 0
yes 5x = 6
yes y/2 = 3
But the variables (like "x" or "y") in Linear Equations do NOT have:
Exponents (like the 2 in x2)
Square roots, cube roots, etc
Examples: These are NOT linear equations:
not y2 − 2 = 0
not 3√x − y = 6
not x3/2 = 16