Sullivan Algebra and Trigonometry: Section 12.1 Systems of Linear Equations Objectives of this Section Solve Systems of Equations by Substitution Solve Systems of Equations by Elimination Identify Inconsistent Systems Express the Solutions of a System of Dependent Equations Solve Systems of 3 Equations Containing 3 Variables Identify Inconsistent Systems of Equations with 3 Variables

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. 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.

An equation in n variables is said to be linear if it is equivalent to an equation of the form are n distinct variables,where are constants, and at least one of the a’s is not zero.

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. Solution y 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. x y

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. x y

Two Algebraic Methods for Solving a System 1. Method of substitution 2. Method of elimination STEP 1: Solve for x in (2)

STEP 2: Substitute for x in (1)

STEP 3: Solve for y STEP 4: Substitute y = -4 into (2) Solution: (3, -4)

STEP 5: Verify solution (1): (2):

Rules for Obtaining an Equivalent System of Equations 1. Interchange any two equations of the system. 2. Multiply (or divide) each side of an equation by the same nonzero constant. 3. Replace any equation in the system by the sum (or difference) of that equation and any other equation in the system.

Multiply (1) by 3 Replace (2) by the sum of (1) and (2) Equation (2) has no solution. System is inconsistent.

Replace (2) by the sum of (1) and (2) Multiply (1) by -2

Replace (3) by the sum of (1) and (3) Multiply (2) by -2 Replace (3) by the sum of (2) and (3)

Multiply (3) by -1/9 Back-substitute; let z = 3 in (2) and solve for y

Back-substitute; let y = -1 and z = 3 in (1) and solve for x. Solution: x = 2, y = -1, z = 3