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 $x_1,\dots ,x_n$ are variables (or unknowns) and $a_1,\dots ,a_n,b$ 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 $x_1,\dots ,x_n$ 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 $\sqrt{2}$.

Quick Quiz

Exercise

Which of the following is equivalent to a linear equation?

1. $2+3x=4$

   Yes.

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

   No.

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

   Yes.