Lesson 1: Solving Linear Systems by Graphing
Module 7: Systems of Linear Equations
Lesson 1 Summary
In this lesson you investigated the following questions:
- What is meant by the solution of a system of equations?
- How is solving a mathematical problem different from verifying a solution to the problem?
In this lesson you learned that a system of linear equations is a set of equations with multiple unknown variables. A linear system can be solved if there are at least as many equations in the system as there are unknowns. The solution to a system of equations identifies the value of each unknown variable such that those values would satisfy every equation in the system. For instance, if one equation describes costs and the other equation describes income, the intersection between the two is often referred to as the break-even point in business. At the break-even point, the business neither realizes a profit nor suffers a loss.
In previous modules you learned how to graph linear functions. In this lesson you built on those graphing skills by solving linear systems in two variables using graphing methods. You used both paper-and-pencil techniques as well as graphing technology to help you graph linear systems. You discovered that the point of intersection of the two lines of a linear system represents the solution to the system. This is due to the fact that the point of intersection lies on both lines, so its coordinates will satisfy both equations.
In addition to learning how to solve linear systems by graphing, you also learned how to verify your solutions. You learned that solving a system is different from verifying a solution. In solving a system, you follow a procedure that leads to the correct solution. On the other hand, verifying is the process of checking potential solutions against the original equations.
In Lessons 3 and 4 you will learn algebraic methods of solving systems of linear equations. At the end of the module, you will compare all of the strategies for solving linear systems including both graphing and algebraic ones.