Monday, April 27, 2026

Systems of Equations

I. Intro to Systems of Equations

A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.

In this section, we will look at systems of linear equations in two variables, which consist of two equations that contain two different variables. For example, consider the following system of linear equations in two variables.

2x + y = 15
3x - y = 5

The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair (4, 7) is the solution to the system of linear equations. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations. Shortly we will investigate methods of finding such a solution if it exists.

2(4) + (7) = 15 True
3(4) - (7) = 5 True

In addition to considering the number of equations and variables, we can categorize systems of linear equations by the number of solutions. 


II. Types of Linear Systems
A consistent system of equations has at least one solution. 

There are two types of consistent system of equations. 
  • A consistent system is considered to be an independent system if it has a single solution, such as the example we just explored. The two lines have different slopes and intersect at one point in the plane. 


  • A consistent system is considered to be a dependent system if the equations have the same slope and the same y-intercepts. In other words, they are the same line. Every point on the line represents a coordinate pair that satisfies the system. Thus, there are an infinite number of solutions.

Another type of system of linear equations is an inconsistent system, which is one in which the equations represent two parallel lines. The lines have the same slope and different y-intercepts. There are no points common to both lines; hence, there is no solution to the system.




III. Determining Whether an Ordered Pair is a Solutions to a System of Equations
In order for an ordered pair to a solution, it needs to be true for both equations. To test, you:

1. Substitute the ordered pair into each equation in the system.
2. Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a solution.

Example
Determine whether the ordered pair (5,1) is a solution to the given system of equations.

𝑥 + 3⁢𝑦 = 8
2⁢𝑥 − 9 = 𝑦

Substitute the ordered pair (5, 1) into both equations.

(5) + 3⁢(1) = 8
8 = 8      True

2⁢(5) − 9 = (1)
1 = 1      True

The ordered pair (5,1) satisfies both equations, so it is the solution to the system.

We can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered pair that satisfies both equations, it is a point on both of the lines and thus the point of intersection of the two lines.



IV. Solving Systems of Equations by Graphing
There are multiple methods of solving systems of linear equations. For a system of linear equations in two variables, we can determine both the type of system and the solution by graphing the system of equations on the same set of axes.


Example
Solve the following system of equations by graphing. Identify the type of system.

2⁢𝑥 + 𝑦 = −8
𝑥 − 𝑦 = −1

Solution:
Solve the first equation for 𝑦.

2⁢𝑥 + 𝑦 = −8
𝑦 = −2⁢𝑥 ⁢−8

Solve the second equation for 𝑦.
𝑥 − 𝑦 = −1
𝑦 = 𝑥 + 1

Graph both equations 



The lines appear to intersect at the point (−3,⁢−2). We can check to make sure that this is the solution to the system by substituting the ordered pair into both equations.

2⁢(−3 ) + (−2) = −8 
−8 = −8               True

(−3) − (−2) = −1
−1 = −1               True

The solution to the system is the ordered pair (−3,⁢−2), so the system is independent.


V. Solving Systems of Equations by Substitution
Solving a linear system in two variables by graphing works well when the solution consists of integer values, but if our solution contains decimals or fractions, it is not the most precise method. We will now look at the substitution method, in which we solve one of the equations for one variable and then substitute the result into the second equation to solve for the second variable.

Algorithm for solving a system of two equations in two variables using substitution

1. Solve one of the two equations for one of the variables in terms of the other.
2. Substitute the expression for this variable into the second equation, then solve for the remaining variable.
3. Substitute that solution into either of the original equations to find the value of the first variable. If possible, write the solution as an ordered pair.
4. Check the solution in both equations.

Example
Solve the following system of equations by substitution.

−𝑥 + 𝑦 = −5
2⁢𝑥 − 5⁢𝑦 = 1

First, we will solve the first equation for 𝑦.
−𝑥 + 𝑦 = −5
𝑦 = 𝑥 ⁢− 5

Now we can substitute the expression 𝑥⁢−5 for 𝑦 in the second equation.

2⁢𝑥 − 5⁢𝑦 = 1
2⁢𝑥 − 5⁢(𝑥 − 5) = 1
2⁢𝑥 − 5⁢𝑥 + 25 = 1
−3⁢𝑥 = −24
𝑥 = 8

Now, we substitute 𝑥 = 8 into the first equation and solve for 𝑦.
−(8) + 𝑦 = −5
𝑦 = 3

Our solution is (8,3).

Check the solution by substituting (8,3) into both equations.
−𝑥 + 𝑦 = −5
−(8) + (3) = −5     True

2⁢𝑥 − 5⁢𝑦 = 1
2⁢(8) − 5⁢(3) = 1     True


VI. Misc. 
A. Other Methods of Solving Systems of Equations
There are other methods of solving systems of equations which are taught in a full algebra class including the addition (elimination) method and matrix methods. 



B. System of Equations Word Problems


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