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The Substitution Method

Introduction

Even relatively untrained algebra learners can find the answer to a simpler system of equations using the guess and check method.  If time permits, graphing the lines is an alternate way to find the solutions. 

It is important to familiarize yourself with the above two methods of solving systems before moving on to tackle more heavy duty methods.  Almost every system of two equations can be solved using the algebraic methods of substitution or elimination.  This lesson demonstrates how to substitute to find the solutions.  Obviously, this method is called the substitution method.

 

Substituting is a concept that has been presented many times since the beginning of algebra. 

 

First, you learned how to substitute for a single variable:

 

Substitution_vis_1 

 

Next, you learned how to substitute for several variables:

Substitution_vis_2 

 

In this lesson, the first step to solving a system of equations is to substitute for an entire side of the equation.

 

Substitution_vis_3

Lesson

When there are many ways to do a particular problem, it makes sense to choose a method that gives you the best chance to get the answer.  If two methods both give you a “best” chance at the correct answer, then choose the one that requires the least amount of work.

 

The difficulty level of a systems of equations may range from “easy” up to “very hard.”  When doing an easy problem, guess and check is a good method to use.  More difficult problems may be nearly impossible to guess, though, so they are best done using another method.  When one of the equations of a system has a variable that can be easily isolated, a good method to use is substitution.

Substituting in a system of equations simply means that you will replace a variable in one of the equations with whatever that variable equals.  The substitution in example 1 is clearly marked so that you can see how the substitution worked.

 

 

Example 1:  Find the solution to the system Substitution_vis_4

Substitution_vis_5

 

So basically the first equation was substituted into the second equation, resulting in an answer of y = 4.  Since there are two variables, you must also find the value of x in your solution.  Since y = 4, you can substitute this value into the equation x = 2y – 1.  Once you have found the value of both variables, rewrite them in a single location… this is your solution.

When both equations already have a variable isolated, then the problem can be done in much the same way.  This type of problem is confusing to many people learning how to solve systems because there is a choice of how to substitute.  You can actually do the problem by substituting either equation into the other one. 

 

 

Example 2:  Find the solution to the system Substitution_vis_6

Substitution_vis_7

 

It would be fine to use the a = 400 – 2b and substitute the (400 – 2b) in for the value of a into the second equation.  The answer works out to be the same this way.  The only advantage that (b + 100) has over (400 – 2b) is that it is a little simpler to work with. 

 

There are times when neither equation has a term that is isolated.  In this case, determine whether one of the equations can be manipulated to isolate one term.  If so, then substitution may be the best method for solving the system.  If it is not easy to isolate a term, then substitution is not a good option and you should consider another method (like elimination.)

 

 

Example 3:  Find the solution to the system Substitution_vis_8

 Substitution_vis_9

 

Since there are so many work steps in these equations, it may be hard to tell if you made a mistake.  Luckily, it is very easy to check your work on these problems… just plug your solution into each equation and see if it works.  It is worth the time and effort needed to check your answer.  Here is a check to the answer of example #3.

 

Substitution_vis_10

 

Another tip is to make sure you leave yourself enough room for all your work.  Efficient students can do about 6 of these problems on one piece of paper.  Avoid the temptation to squeeze in work or leave out steps in order to save paper.  After you have spent all that time doing the problem, be sure you got it right by checking to see if your answer is right.  If not, try the problem again.  Only after you have tried a second and third time and still are coming up with a solution that doesn’t work should you consider getting a second opinion on how to do the problem.  Remember that you must think for yourself in order to improve in mathematics and feel really confident about your ability.  Don’t rob yourself of that opportunity by being impatient as you complete your assignment.

 

Try It

Find the solution to each system of equations.

1)  Substitution_vis_11

2)  Substitution_vis_12

3)  Substitution_vis_13

4)  Substitution_vis_14 

 

 

Scroll Down for Answers…

 

 

 

 

 

 

Answers:

1)  s = 15, t = 4

2)  p = 30, q = 12

3)  m = 6, n = 9

4)  f = 6, g = 3

 

 

 

 

Related Links:

Looking for a different lesson on systems of equations?  Try the links below.

Related Lessons

 

Before attempting to learn systems of equations, one should be comfortable solving a variety of (single) equations. 

Equation Lessons

 

Looking for something else?  Explore our menu of general math or algebra lessons.

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