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.
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.
\(x - 2y + 3z = 4\) 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 \(\sqrt{2}\).
Which of the following is equivalent to a linear equation?
\(2+3x = 4\)
\((3x-1)(2x+1) = 0\)
\(2(1-x)x+2x^2 = 3x-1\)