How do you find the solutions of systems?

Solution is a word that we frequently use in math, but it can mean different things depending on its context. In general, however, a solution is a value or set of values that make equations true. Although the idea of truth may seem like something more relevant to disciplines such as science and philosophy than math, we’re seeking truth when we look for solutions to systems of equations.

What is a Solution to a System of Equations?

To figure out what the solution to a system of equations is, let’s start by looking at some equations and their solutions.

What do the two equations and their solutions have in common? The solutions make the equations true. When s=9, then 5+4=s. When n=2, then n+7=9.

A system of equations involves two or more equations. Each of the equations must have at least two variables, for example, x and y.

To review what a system of equations is, check out our post: Writing Systems of Equations. 

The solution set to a system of equations will be the coordinates of the ordered pair(s) that satisfy all equations in the system. In other words, those values of x and y will make the equations true. Accordingly, when a system of equations is graphed, the solution will be all points of intersection of the graphs.

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Number of Solutions to a System of Equations

The number of solutions to a system of equations depends on the equations themselves. Systems can have one solution, multiple solutions, infinitely many solutions, or even no solution.

Systems of linear equations are categorized by how many solutions they have. There are two main categories of systems of equations:

• An inconsistent system, which has no solutions
• A consistent system, which has one or more solutions

Consistent systems can be further subdivided into:

• An independent system, which has exactly one solution
• A dependent system, which has an infinite number of solutions

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Systems of Equations with No Solution (Example)

Let’s start by looking at inconsistent systems, what we call it when there are no solutions to systems of equations. We know that parallel lines never intersect. Therefore, if our system of equations consists of two or more parallel lines, there will be no places where the graphs of the lines intersect, and thus there will be no solutions. 

For a review of parallel lines, check out our review guide Parallel and Perpendicular Lines.

The lines y=x+3 and y=x-2 are parallel. Their graphs are shown to the right.

From the graph, we can see that the lines never intersect, and thus there are no solutions to this system of equations. So although each linear equation in the system has an infinite number of solutions, the system of equations consisting of both these linear equations has no solutions. We can classify the system as inconsistent.

Systems of nonlinear equations can also have no solutions. Below is the graph of a system of two quadratic equations that never intersect. Together, these equations make up a system with no solutions.

What about systems with more than two equations? Solutions must satisfy all equations in the system. Graphically, only points of intersection of all graphs in the system count as solutions. 

For example, the graph below shows a system of three equations: two parallel lines and one line that intersects the parallel lines. This system of equations has no solution because there is no place where all three lines intersect with each other simultaneously. 

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System of Equations with Infinite Solutions (Example) 

Next, let’s go to the opposite extreme and examine systems of equations that are both consistent and dependent, which occurs when there are infinite solutions to systems of equations. Graphically, we’re looking for a system of equations that intersects at an infinite number of points. How can that happen? It happens whenever the two equations are actually the same equation. 

For example, consider the system:

y=x+3

x=y-3

Although the second equation is not written in slope-intercept form, we can see that the equation has the same slope, 1, and the same y-intercept, 3, that y=x+3 has.

For a quick recap of forms of linear equations, check out our blog post, Slope-Intercept Form.

This fact that the two equations are identical will become obvious if we rewrite x=y-3 in slope-intercept form:

x=y-3

x{\color{red}{-y}}=y-3{\color{red}{-y}}

x-y=-3

x-y{\color{red}{-x}}=-3{\color{red}{-x}}

-y=-x+-3

{\color{red}{(-1)}}(-y)={\color{red}{(-1)}}(-x+-3)

y=x+3

So, we can also write this system as:

y=x+3

y=x+3

The graphs of these two equations are shown to the right.

As can be seen from the graph, these equations represent the same line on the coordinate plane. Since all points on one line are also on the other line, there are an infinite number of solutions to this system.

Note that this does not mean that every set of coordinates is a solution to this equation. For instance, the coordinates (0,5) are not a solution to the system of equations since that point is not on the lines. However, since a line stretches to infinity in both directions, there exists an infinite number of points on each line, and thus an infinite number of solutions to the system of equations. 

Other systems of equations that are both consistent and dependent include any combination of equations, such as equations for circles, parabolas, and other figures, where the graphs of the equations fully overlap.

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One Solution System of Equations (Example) 

How many solutions to systems of equations are there for a system that is both independent and consistent? This kind of system has exactly one solution, which means it has one set of values that makes all equations in the system true.

The linear equations in the system below fit this description: 

y=x+3

 y=-x-1 

The lines that represent these equations on the coordinate plane will intersect at exactly one point, as shown below.

Since these lines intersect at the point (-2,1), that point is the only solution to the system. We can use algebra to confirm that the coordinates (\color{red}{-2},\color{blue}{1}) make each equation into a true statement. 

Let’s start with the first equation:

y=x+3

{\color{blue}{1}}={\color{red}{-2}}+3

1=1

And now let’s test the second equation:

y=-x-3

{\color{blue}{1}}=-({\color{red}{-2}})-1

1=2-1

1=1

Note that we could have determined that this system of equations had one solution just from looking at the equations themselves if we notice that they have different slopes. The slope of y=x+3 is 1, and the slope of  y=-x-1 is -1. Lines with different slopes are never parallel, and lines that are co-planer but not parallel will always intersect at exactly one point.

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How to Find the Solution to a System of Equations

There are multiple methods for solving systems of equations, including: 

• Graphing each equation on the coordinate plane to find the points of intersection. Tools like Desmos and Geogebra offer online graphing calculators to help in this process.
• Using algebraic strategies like substitution or elimination to find the values for the variables that make all equations in the system true. 

Each of these methods can help us find not only the number of solutions to systems of equations, but also what those solutions are.

When we find the solution or solutions to a system of equations, we can substitute the values back into the equations to make sure they are correct.

How do you find the solution set of a system of equations?

How do you know how many solutions a system has?

What are the 3 methods to solving systems?