Mat 350 Notes1.1 Systems of linear equationsSection summaryIntroductory algebra students study linear equations in twovariables. However, a linear equation can have more than just twovariables. Linear equations with multiple variables are used in manydisciplines. Common applications include analyzing data, creatingpredictive models, and optimizing processes given a set ofconstraints. Often, a real-world scenario can be modeled by severallinear equations, which are referred to as a system of linearequations.In this section, students will:Identify a linear equation.Determine whether ann-tuple is a solution to a system oflinear equations.Identify the type of linear system geometrically.Set up a system of linear equations that models a real-lifesituation.Linear equationsAlinear equationis an equation of the following forma1x1+a2x2+⋯+anxn=bwhereaiis acoefficient,xiis avariablefor all integersi=1,…,n,andbis a constant.The graph of a linear equation in two dimensions is a line and thegraph of a linear equation in three dimensions is a plane, as shownin figures (a) and (b) below. However, graphing a linear equationwhenn>3is not possible.Figure 1.1.1: Graphs of linear equations in two and threedimensions.
Feedback?Systems of linear equationsAsystem of linear equations, orlinear system, is a collection ofone or more linear equations involving the same variables. A systemofmequations innvariables has the forma11x1+a12x2+…+a1nxn=b1a21x1+a22x2+…+a2nxn=b2⋮am1x1+am2x2+…+amnxn=bnwhereaijis the coefficient ofxjandbiis the constant term in theithequation fori=1,…,mandj=1,…,n.The following two equations form a system of linear equationsinx1andx2.x1+x2=2x1−x2=4Asolutionof a system of linear equations withnvariables is anordered list ofnelements, called ann-tuple, written as(x1,x2,…,xn),that satisfies every equation in the system. To determine whetherann-tuple is a solution to a system, each element is substituted intothe corresponding variable. If the result is true forallequations,then then-tuple is a solution.Geometric representation of a solution to a system of linearequations
Geometrically, a solution to a system of linear equations is ann-tuple that identifies the point where the hyperplanes intersect. Thus,a system can have one solution, no solution, or infinitely manysolutions.The following terms are used to classify systems based on theexistence and uniqueness of a solution:Aconsistent systemhas one solution or infinitely manysolutions.Anindependent systemis a consistent system with onesolution.Adependent systemis a consistent system with infinitelymany solutions.Aninconsistent systemhas no solution.Applications of linear systemsMany real-life situations can be modeled by a linear equation.