Mathematics 10C Module 7 Lesson 5

Module 7: Systems of Linear Equations

Lesson 5 Summary

In this lesson you investigated the following questions:

  • Why can linear systems have different numbers of solutions?
  • How do the equations of a linear system indicate the number of solutions of that system?

Up until this lesson, you focused on systems that had only one solution or those systems whose graphs contained only one point of intersection. In this lesson you learned that a system of linear equations with two variables can also have no solutions or an infinite number of solutions. You discovered this by graphing each of the different ways that two lines can be oriented relative to each other on a coordinate plane. If two lines are parallel, then they will have no solutions because there will not be a point of intersection. On the other hand, if two lines are coincident, then they will have an infinitely many points of intersection and, therefore, infinitely many solutions.

Besides analyzing graphs, you also analyzed the equations of systems for clues about the number of solutions for the systems. You discovered that, by determining the slopes and the y-intercepts of each equation in the system, you could identify the number of solutions. If the equations have a common slope as well as a common y-intercept, then the system has an infinite number of solutions. A system whose equations have only a common slope, but differing y-intercepts will have no solution since the lines will be parallel. Finally, a system whose equations do not share the same slope will have only one solution.

Another way of identifying the number of solutions of a linear system is by analyzing the coefficients of each equation in standard form. For example, if one equation is a multiple of the other equation, then there are infinitely many solutions. If the left side of one equation is not a multiple of the left side of the other equation, then there is only one solution. All other cases represent those systems with no solutions.

In the final lesson of this module, you will work with others to solve problems modelled by systems of linear equations. You will revisit all of the strategies that you have learned to solve linear systems. You will practise modelling systems and solving them based on the selection of the most appropriate strategy for each problem.

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