If *a*, *b*, and *c *are real numbers, the graph of an equation of the form

*ax *+ *by *= *c*

is a straight line (if *a *and *b *are not both zero), so such an equation is called a *linear *equation in the variables *x *and *y*. However, it is often convenient to write the variables as *x*1, *x*2, …, *x**n*, particularly when more than two variables are involved.

An equation of the form

*a*1*x*1 + *a*2*x*2 + _ + *a**n**x**n *= *b*

is called a **linear equation **in the *n *variables *x*1, *x*2, …, *xn*. Here *a*1, *a*2, …, *an *denote real numbers (called the **coefficients **of *x*1, *x*2, …, *xn*, respectively) and *b *is also a number (called the **constant term **of the equation). A finite collection of linear equations in the variables *x*1, *x*2, …, *xn *is called a **system of linear equations **in these variables.

Given a linear equation *a*1*x*1 + *a*2*x*2 + _ + *a**n**x**n *= *b*, a sequence *s*1, *s*2, …, *s**n *of *n *numbers is called a **solution **to the equation if

*a*1*s*1 + *a*2*s*2 + _ + *a**n**s**n *= *b*

that is, if the equation is satisfied when the substitutions *x*1 = *s*1, *x*2 = *s*2, …, *x**n *= *s**n *are made. A sequence of numbers is called **a solution to a system **of equations if it is a solution to every equation in the system.