The following are examples of equations in one variable (or unknown) \(x\):

\(3x - 1 = 2\)

\(2x^2 - 3x + 1 = 0\)

\(\sin(x) + e^{x^2} = 5\)

For each of these,
the question is to find a value
that we can assign to \(x\) so that the equality is satisfied.

It is not hard to see that assigning 1 to \(x\) satisfies the first equation.
For the second equation, one can use the quadratic
formula to find all the solutions. The third equation is a bit complicated
and there is no known method for solving it exactly.

Definition of a linear equation

A linear equation is an equation of the form:
\[\sum_{i=1}^n a_i x_i = b\]
where \(x_1,\ldots, x_n\) are variables (or unknowns)
and \(a_1,\ldots, a_n, b\) are
constants.
The contant \(a_i\) is called the coefficient of the variable
\(x_i\).
A solution is an assignment of values to the variables
\(x_1,\ldots,x_n\) such that the left-hand side is equal to the right-hand
side.

A linear equation is normally defined over a field;
i.e. the constants are elements of a field and the values
we solve for the variables are from the same field.

Note that equation 1 above is not quite in this form yet. But it can
be turned into this form by adding \(1\) to both sides of the equation
to obtain the equivalent \(3x = 3\). (Two equations are said to be
equivalent if they have the same solutions.)

Equations that are not linear are called nonlinear equation.
Hence, equations 2 and 3 above are both nonlinear equations.

Examples

\(x - 2y + 3z = 4\) is a linear equation in the variables \(x,y,z\).
Here, the coefficient of \(x\) is 1.
One solution (there are many others) is given by
\(x = 3\), \(y = 1\), \(z = 1\).

\(x_1 - \pi x_2 + 3x_3 - \sqrt{2} x_4 = 0\) is a linear equation in
the variables \(x_1,x_2,x_3,x_4\). Here, the coefficient of \(x_2\) is
\(\pi\) and the coefficient of \(x_4\) is \(\sqrt{2}\).