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What is a System of Equations?

Introduction

Most algebra learners are familiar with the term “equation” as well as the rules and methods for finding the solution to an equation.  A system of equations is a more complex problem that involves finding a solution to two separate and independent equations.  Systems of equations are often connected with curly braces to show that the two equations go together.  There are several different methods that can be used to find the solution to a system of equations, and the first method to become familiar with is the method of “guess and check.”  This method is shown below for an equation as well as a system of equations.

What_is_a_system_vis_1

 

The solutions above can be found by guessing the values of the variables until you find the one that works.  The equation is easier because there is only one variable to guess.  In the system of equations, there is a double solution so there are more combinations of variables to guess.    

 

Lesson

Finding a pair of solutions that works is the goal when dealing with a system of equations.  A good way to get used to systems is to look at some easier problems that can be done by putting forth an (educated) guess for each of the two variables, then seeing if the guess is correct as a solution to the system.

 

 

 

Example 1:  Find the solution to the system  What_is_a_system_vis_2

What_is_a_system_vis_3 

 

In each of the guesses above, the two numbers were chosen because they worked for the first equation.  The solution is the one that works for the first equation and also works in the second equation.  You could have also used the second equation to choose possible solutions, then checked the numbers in the first equation to see if they worked.

Example 2 is a system of equations that you can also do using the “guess and check” method.  See if you can find the solution.

  

 

Example 2Find the solution to the system  What_is_a_system_vis_4

What_is_a_system_vis_5What_is_a_system_vis_6

 

 

 

If the equations are simple enough and you have the time, guess and check can be a good way to find the solution of a system.  When equations get more involved there are other methods that use the rules of mathematics to help you find the solution.

Try It

Find the solution to each system of equations.

1)  What_is_a_system_vis_7

 2)  What_is_a_system_vis_8

3)  What_is_a_system_vis_9

4)  What_is_a_system_vis_10 

 

 

Scroll Down for Answers…

 

 

 

 

 

 

 

Answers:

1)  c = 6, d = 4

2)  e = 15, f = 12

3)  g = -2, h = 10

4)  i = 9, j = 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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