Examples of One Solution Equations, Zero, and Infinite

Solving algebraic equations is one of the key skills that you will develop in your studies of algebra. One of the first types of equations that you will learn to solve are linear equations and linear systems of equations.

In my experience, some of my students have been confused over whether an equation has one solution, infinite solutions, or zero solutions.

I have put together this list of examples of one solution equations, zero solution equations, and infinite solutions equations to help you learn the difference!

How Do you Tell if an Equation has One Solution, No solution, or Infinite Solutions?

When you are working with systems of linear equations there are three possible types of solutions:

• One solution: the non-parallel lines intersect at a single point.
• No solution: the lines do not intersect because they are parallel.
• Infinite solutions: the lines intersect everywhere because they fall on top of one another.

So how do you tell if an equation has one solution, no solution, or infinite solutions? The trick is to look at the coefficients and constant terms of each given equation in the system of equations. 

• If the equations are simplified and the result is a true statement, there will be one single point of intersection. A single linear equation will also only ever have one solution
• If the equations are simplified and the result is a false statement, there will be no solution.
• If the coefficients in each line are multiples of one another, there will be infinite intersections.

Let’s take a look at some examples of each type of system so that you can develop an understanding of each case.

Examples of One Solution Equations

Let’s begin by looking at a few examples of one solution equations. 

Example 1 – Consider the following single equation:

$$x – 4=9$$

One-step equations like this example are solvable by performing a single step. We can use algebraic strategies to determine the value of the unknown variable x. We do this by performing the opposite operation.

Since we have x subtracted by 4, adding 4 to both sides of the equation will allow us to move all non-variable terms to the right-hand side of the equation.

$$ \begin{split} x – 4&=9 \\ \\  x -4 + 4 &= 9 + 4 \\ \\ x&=13\end{split} $$

Therefore, x=13 is the value of the variable. This is also the only solution to this linear equation!

Example 2 – Consider the following system of linear equations:

$$ \begin{split} 2x+3y&=13 \\ 4x-y&=5 \end{split} $$

This system contains two linear equations, each written in standard form. Notice that the coefficients and constant terms are not multiples of one another.

We can apply substitution or elimination to solve this system. 

If we choose substitution, we can rearrange the second equation to solve for y. This results in the slope-intercept form equation \(y=4x-5\). We can substitute this new expression into the first equation to solve for x.

Remember to apply inverse operations to isolate the value of x.

$$ \begin{split} 2x+3(4x-5) &=13 \\ \\  2x+12x-15 &=13 \\ \\ 14x &=28 \\ \\ x&=2\end{split} $$

We then substitute x=2 into either original equation to find the value of y.

$$ \begin{split} 2(2)+3y&=13 \\ \\ 4+3y&=13 \\ \\  3y&=9 \\ \\ \frac{3y}{3} &= \frac{9}{3} \\ \\ y&=3 \end{split} $$

Since we are working with a system of equations, the solution of the equation is a single point rather than a single x-value like the previous example. In this case, there is one solution to this linear system of equations located at the point (2,3).

Examples of Linear Equations with No Solution

Example 1 – Consider the following pair of equations

$$ \begin{split} y&=2x+4 \\ y&=2x+6 \end{split} $$

At first glance, we can see that each of these slope-intercept form equations have the same slope. In each case, the slope of the line is 2. Since the equations have different y-intercepts, we know that the lines will be parallel to one another.

Let’s take a look at what happens when we set the equations equal to one another to determine the solution. Let’s attempt to bring all variable terms to the left-hand side and all constant terms to the right side of the equation:

$$ \begin{split} 2x+4&=2x+6 \\ \\ 0 &= 2 \end{split} $$

Notice that the 2x terms cancelled when we used algebra to gather all x terms on one side of the equation. The result is a false statement.

Therefore, we can say that this system of linear equations has no solution.

Example 2 – Consider the following pair of equations

$$ \begin{split} 5x-y&=2 \\ 10x-2y&=6 \end{split} $$

We may not be able to determine whether the lines are parallel just by looking at them. But, we can apply a little bit of algebra to rewrite these standard form equations in slope-intercept form to see what we are working with.

The first equation can be written as:

$$ \begin{split} 5x-y&=2 \\ \\ -y&=-5x+2 \\ \\ y&=5x-2 \end{split} $$

The second equation can be written as:

$$ \begin{split} 10x-2y&=6 \\ \\ -2y&=-10x+6 \\ \\ y&=5x-3 \end{split} $$

Notice that in each case the slope of the line is 5. As a result, if we set these equations equal to one another, we will see that there is no solution.

Example of Linear Equations with an Infinite Number of Solutions

Linear equations with an infinite number of solutions are easy to identify if you know what to look for. Essentially, the equations that make up the linear system will be the same line written in different ways.

Example: Consider the following system of linear equations.

$$ \begin{split} 2x+3y&= 5 \\ 4x+6y&=10 \\ \\ \end{split} $$

The first step I always take when I am solving a linear system of equations is checking whether the second line is a multiple of the first line. 

Compare the first equation to the second equation. Notice that the second line’s coefficients and constant term are twice the size of the first line.

In this case, if we multiply the first line by 2, we produce the second line. This tells us that these two equations actually represent the same line!

We can try to solve this system using the elimination method by first multiplying the first line by 2.

$$ \begin{split} 2(2x+3y)&= 2(5) \\ 4x+6y&=10 \\ \\ \end{split} $$

However, when we go to eliminate a variable, you will see that both variables on the left hand side and the constant on the right hand side cancel out.

$$ \begin{split} 4x+6y&=10 \ \\ 4x+6y&=10 \\ \\ 0&= 0 \end{split} $$

The result is the statement 0=0. This tells us that every point on one line also falls on the other, creating infinite solutions!

Practice with Linear Equations

Many people assume that linear equations are simple examples of one solution equations. However, as we have seen here, linear equations can be much more complex! 

The key to understanding each of the scenarios is to practice recognizing and working with them. The more you practice, the easier it will be to understand and identify the type of solution a linear system has.

I hope that these examples of different solutions of linear equations has helped you understand the different possible solutions that you can encounter when working with linear equations.

If you are looking for more practice with linear equations, check out this linear equations word problems worksheet with solutions to start mastering this topic today!

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