## Examples of single-variable equations

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

1. $$3x - 1 = 2$$

2. $$2x^2 - 3x + 1 = 0$$

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

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

1. $$x - 2y + 3z = 4$$ 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 $$x = 3$$, $$y = 1$$, $$z = 1$$.

2. $$x_1 - \pi x_2 + 3x_3 - \sqrt{2} x_4 = 0$$ 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 $$\sqrt{2}$$.

## Exercise

Which of the following is equivalent to a linear equation?

1. $$2+3x = 4$$

2. $$(3x-1)(2x+1) = 0$$

3. $$2(1-x)x+2x^2 = 3x-1$$