Lesson 3: Solving Linear Systems by Substitution: Summary

Module 7: Lesson 3

Module 7: Systems of Linear Equations

Lesson 3 Summary

In this lesson you investigated the following questions:

How do you decide which variable to isolate? Does it matter?
Why would you change a mathematical equation into an equivalent expression?

In this lesson you learned how to solve linear systems using the substitution method. Like a recipe, where certain substituted ingredients can work just as well as another, the method of substitution can yield a correct solution just as well as the graphing method. In fact, substitution maintains an advantage over graphing. Sometimes, graphing may not yield an exact solution whereas substitution can.

You learned that when you employ substitution, you need to choose a variable to isolate. While it does not matter what variable you isolate, you may have discovered that it makes the most sense to choose the equation with the variable that is most easily isolated. This is most often the case when there is a variable with a coefficient equal to 1 or –1. In instances where no such term exists, you can still isolate any variable; however, the resulting expression may involve rational terms.

When you do encounter linear systems with rational coefficients, as you did in this lesson, you can multiply each term by a constant in order to obtain an equivalent linear system with integral coefficients. Depending on the constant that you choose, doing so could help you more easily isolate a variable.

In this lesson you engaged in several inquiry activities, including group work and applet-based exercises. You determined solutions and verified them according to previously learned techniques.

In the next lesson you will learn a second algebraic method that can be used to solve linear systems that is not amenable to substitution or graphing. You will see that there are certain ways of manipulating linear systems that do not change the solution to a problem. This new method, along with substitution, will serve as a solid foundation for you to solve any linear system.