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Chapter 1 Systems of Linear Equations (LE)
One of the fundamental lessons of calculus is that many complicated problems can be solved by taking linear approximations to complicated functions. This often produces several linear equations for which we want to know when they are simultaneously true; that is, we would like to solve this
system of linear equations. This chapter centers around that fundamental question, building off of your intuition for what happens in the simplest case of two equations in two variables. Geometrically, this corresponds to finding the intersection of two lines in the plane. As we introduce more variables, we quickly lose our ability to visualize this process geometrically, so we will introduce a tool called a
matrix to help us solve these.
Motivating Question. How can we solve systems of linear equations?
Learning Outcomes By the end of this chapter, you should be able to...
Translate back and forth between a system of linear equations, a vector equation, and the corresponding augmented matrix.
Explain why a matrix isn’t in reduced row echelon form, and put a matrix in reduced row echelon form.
Determine the number of solutions for a system of linear equations or a vector equation.
Compute the solution set for a system of linear equations or a vector equation with infinitely many solutions.
Readiness Assurance.
Before beginning this chapter, you should be able to...
Determine if a system to a two-variable system of linear equations will have zero, one, or infinitely-many solutions by graphing.
Find the unique solution to a two-variable system of linear equations by back-substitution.
Describe sets using set-builder notation, and check if an element is a member of a set described by set-builder notation.
