10.1 SYSTEMS OF LINEAR EQUATIONS: SUBTRACTION, ELIMINATION

A system of equations is a collection of two or more equations, each containing one or more variables. A solution of a system of equations consists of values for the variables that reduce each equation of the system to a true statement. A solution of a system of equations consists of values for the variables that reduce each equation of the system to a true statement. To solve a system of equations means to find all solutions of the system. To solve a system of equations means to find all solutions of the system. When a system of equations has at least one solution, it is said to be consistent; otherwise it is called inconsistent. When a system of equations has at least one solution, it is said to be consistent; otherwise it is called inconsistent.

If the graph of the lines in a system of two linear equations in two variables intersect, then the system of equations has one solution, given by the point of intersection. The system is consistent and the equations are independent. y Solution x

If the graph of the lines in a system of two linear equations in two variables are parallel, then the system of equations has no solution, because the lines never intersect. The system is inconsistent. y x

If the graph of the lines in a system of two linear equations in two variables are coincident, then the system of equations has infinitely many solutions, represented by the totality of points on the line. The system is consistent and dependent. y x

Two Algebraic Methods for Solving a System 1. Method of substitution 2. Method of elimination