A linear equation with two variables, say x and y, is of the form ax + by = c, where a, b and c are constants. Here, a and b are called coefficients. The coefficients should be nonzero for the equation to have two variables. Our aim in this unit is to find solutions that satisfy not just one equation but two linear equations at the same time.

Two linear equations in two variables taken together are called simultaneous linear equations.

Definition

A simultaneous linear equations are system of two linear equations in two variables, say x and y, given as a₁x + b₁y = c₁ and a₂x + b₂y = c₂, where a_i, b_i and c_i are constants. This is usually written as

Solving a simultaneous equation means finding pair of values for the unknown variables x and y which satisfy both equations at the same time. We write such a pair of values (solutions) as an ordered pair (x,y) . This is stated in the following definition.

Definition

A solution to a simultaneous system of two equations in two variables, say x and y, is an ordered pair (x,y) that satisfies both equations. The set of all such solutions is called truth set or solution set.