In addition to the reflection activities below, re-read Section 1.2 Terminology and notation. Be sure you understand each of the new definitions introduced in this chapter, and spend some time committing them to memory.
1.Testing system solutions.
Explain how to test whether a specific set of variable values is a solution to a system of equations.
2.Representing system solutions.
Explain how parametric equations can represent many different solutions to a system of equations.
3.Representing systems via matrices.
Explain what the entries in an augmented matrix represent.
4.The algebra of row operations.
Explain the connections between row operations in an augmented matrix and algebraic manipulations of equations in a system.
5.Number of equations versus number of unknowns.
In your previous mathematical studies, you may have heard the “rule of thumb” that to solve for \(n\) variables you need \(n\) equations.
(a)
Use lines in the Cartesian plane to explore whether this “rule of thumb” is accurate for linear equations in two variables.
(i)
Does a system of one linear equation in two unknowns have solutions?
(ii)
Does a system of two linear equation in two unknowns always have a solution?
(iii)
Can a system of three linear equations in two unknowns have a solution?
(b)
Use planes in three-dimensional space to explore whether this “rule of thumb” is accurate for linear equations in three variables. (You may wish to use pieces of papers as props to represent different planes.)
(i)
Does a system of one linear equation in three unknowns have solutions?
(ii)
Can a system of two linear equations in three unknowns have solutions?
(iii)
Does a system of three linear equations in three unknowns always have a solution?
(iv)
Can a system of four linear equations in three unknowns have a solution?
Based on your reflections above, rewrite the “rule of thumb” to more accurately express the intended meaning.
