Systems Of Linear Equations With Infinite Solutions (3 Ways To Tell)

When working with systems of linear equations, we often see a single solution or no solution at all.  However, it is also possible that a linear system will have infinitely many solutions.

So, when does a system of linear equations have infinite solutions?  A system of two linear equations in two variables has infinite solutions if the two lines are the same. From an algebra standpoint, we get an equation that is always true if we solve the system. Visually, the lines have the same slope and same y-intercept (they intersect at every point on the line).

Of course, a system of three equations in three variables has infinite solutions if the planes intersect in an entire line (or an entire plane if all 3 equations are equivalent).

In this article, we’ll talk about how you can tell that a system of linear equations has infinite solutions.  We’ll also look at some examples of linear systems with infinite solutions in 2 variables and in 3 variables.

Let’s begin.

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Systems Of Linear Equations With Infinite Solutions

A system of linear equations can have infinite solutions if the equations are equivalent.  This means that one of the equations is a multiple of the other.

It also means that every point on the line satisfies all of the equations at the same time.

The image below summarizes the 3 possible cases for the solutions for a system of 2 linear equations in 2 variables.

A system of equations in 2, 3, or more variables can have infinite solutions.  We’ll start with linear equations in 2 variables with infinite solution.

When Does A Linear System Have Infinite Solutions? (System Of Linear Equations In 2 Variables)

There are a few ways to tell when a linear system in two variables has infinite solutions:

• Solve the system – if you solve the system and get an equation that is always true, regardless of variable value (such as 1 = 1), then there are infinite solutions.
• Look at the graph – if the two lines are the same (they overlap, or intersect everywhere on the line), then there are infinite solutions to the system.
• Look at the slope and y-intercept – solve both equations for y to get slope-intercept form, y = mx + b.  If the two equations have the same slope and the same y-intercept, then the lines are equivalent and there are infinite solutions (you can get a refresher on how to tell when two lines are parallel in my article here).

We’ll look at some examples of each case, starting with solving the system.

Solving A Linear System With Infinite Solutions

When we attempt to solve a linear system with infinite solutions, we will get an equation that is always true as a result.  For example, after we simplify and combine like terms, we will get something like 1 = 1 or 5 = 5.

Let’s take a look at some examples to see how this can happen.

Example 1: Using Elimination To Show A Linear System Has Infinite Solutions

Let’s say we want to solve the following system of linear equations:

• 2x + 4y = 3
• -6x – 12y = -9

We will use elimination to solve.  Let’s try to eliminate the “x” variable.

We begin by multiplying the first equation by 3 to get:

• 3(2x + 4y) = 3(3)  [multiply the first equation by 3 on both sides]
• 6x + 12y = 9  [distribute the 3 through parentheses]

Now we add this modified equation to the second one:

6x + 12y = 9

+

-6x – 12y = -9

___________

0x + y = 0

This implies 0 = 0, which is always true – regardless of the values of x or y we choose.  This means that both equations represent the same line.

It also means that every point on that line is a solution to this linear system.  So, the system has infinite solutions.

The graph below shows the line resulting from both of the equations in this system.

Example 2: Using Substitution To Show A Linear System Has Infinite Solutions

Let’s say we want to solve the following system of linear equations:

• y = 2x + 5
• 10x – 5y = -25

We will use substitution to solve.  We’ll substitute the y from the first equation into the y in the second equation:

• 10x – 5y = -25  [start with the second equation]
• 10x – 5(2x + 5) = -25  [substitute y = 2x + 5 from the first equation]
• 10x – 10x – 25 = -25  [distribute 5 through parentheses]
• 0x = 0  [combine like terms on opposite sides]

This implies 0 = 0, which is always true – regardless of the values of x or y we choose.  This means that both equations represent the same line.

It also means that every point on that line is a solution to this linear system.  So, the system has infinite solutions.

The graph below shows the line resulting from both of the equations in this system.

Looking At The Graph Of A Linear System With Infinite Solutions

When we graph a linear system with infinite solutions, we will get two lines that overlap.  That is, they intersect at every point on the line, since the two equations are equivalent and give us the same line.