Equations Reducible to a Pair of Linear Equations in Two Variables An equation in the form of $$ax + by = c$$ is called a linear equation in two variables....
Equations Reducible to a Pair of Linear Equations in Two Variables
An equation in the form of is called a linear equation in two variables. A system of linear equations in two variables is a set of two linear equations in the same two variables.
An equation is reducible to a pair of linear equations if it can be transformed into a system of two linear equations in the same two variables. This means that it can be rearranged into the form of for two variables.
A linear equation in two variables is reducible to a pair of linear equations if and only if it can be expressed in the form of where A, B, and C are constants.
Examples:
is reducible to a pair of linear equations because it can be rearranged into the form of .
is not reducible to a pair of linear equations because it is not in the form of $$ax + by = c$.
Properties of Equations Reducible to a Pair of Linear Equations:
An equation is reducible to a pair of linear equations if and only if it has a solution.
A system of two linear equations in two variables is equivalent to the reduced equation if and only if the two equations are linearly independent.
A system of two linear equations in two variables is consistent if and only if the determinant of the coefficient matrix is non-zero