Question

In this lesson, you are learning about systems of equations and three methods for solving them: graphing, substitution, and addition. Compare and contrast the three methods by discussing the following:

Is one method easier than the others for all systems, or does it depend on the system?

If it depends on the system, how could you tell before you begin solving the system which solution method would be the most efficient?

How is it possible to tell by inspection (look at it) whether a system of linear equations has one, zero, or an infinite number of solutions?

Answer 1

There indeed are three methods for solving a given system of equations viz. graphing, substitution and addition (with a view to eliminate a variable).

When there are only 2 variables, and the given system is not linear or quadratic, graphing is the most suitable and easiest method to locate the zeros which are the points where the graph crosses the X-Axis.

If on preliminary scrutiny, we are able to locate the zeros by the rational/integral roots theorem or otherwise, ( if f(a) = 0, then a is a zero of f(x) = 0), and given system is linear or quadratic, then the other methods are also relatively easy to use. Given a linear system of 2 variables, both the substitution and addition methods are very easy to use in solving the equations. When there is a linear system of more than 2 variables, then also, these 2 systems are very easy to use and are preferred over graphing. However, a fourth system i.e. row operations /Gaussian elimination applied on the augmented/coefficient matrix (depending upon whether the system is non-homogeneous or homogeneous) is the fastest and easiest method. The relative ease of a method depends on the system.

A homogeneous system of linear equations always has a trivial solution i.e. the system is always consistent. When there is at least one free variable, i.e. if some variables can be expressed in terms of one or more other variables, then there are infinite solutions. When there are no free variables, then the system is either inconsistent (i.e. has no solution) or a unique (one only) solution.