Systems of equations are a set of two or more equations with two or more variables. To find the solution to a system of equations, we have to solve the equations simultaneously. However, not all systems of equations have a solution. We can have systems with no solutions, or even systems with infinite solutions.
Here, we will learn how to determine when a system of equations has one solution, no solutions, or infinite solutions.
Systems of equations and their solutions
Systems of equations are sets of two or more equations that have to be solved simultaneously. A solution to the system of equations has to satisfy all the equations in the system at the same time.
In the case of 2×2 systems of equations, a solution to the system is an ordered pair $latex (x,~y)$ that makes both equations true. A solution is what the equations have in common, it is the place where their lines intersect.
If an ordered pair only satisfies one of the equations in the system, but not the other, then it is not a solution to the system of equations.
Depending on whether the system of equations has solutions or not, we can have two types of systems of equations, consistent systems, and inconsistent systems.
- A consistent system is a system that has at least one solution.
- An inconsistent system is a system that has no solutions.
There are three possible results that we can find when working with systems of linear equations:
1. One solution
2. No solution
3. Infinite solutions.
Systems of equations with only one solution
For systems of two equations in two variables, an ordered pair $latex (x,~y)$ is a solution to the system only if it satisfies both equations. That is, by using the ordered pair in each of the equations, both equations must be true.
A system of linear equations of two or more equations with two or more variables will have only one solution when the equations of the system are different from each other and are not parallel. The details of these cases are shown below.
When we obtain a unique solution to the system of equations as the final answer, the system is consistent. The following graph shows a system of two equations with two unknowns that has a solution:
We can see that the graphs of the equations of the system intersect at only one point.
Therefore, in the case of systems of linear equations, one way to determine whether the system will have a unique solution is to draw simple graphs of the equations and see if the lines intersect at a single point.
EXAMPLES
Examples of systems of equations with one solution:
$latex \begin{cases}x+2y=10 \\ 2x-y=5 \end{cases}$, solution: $latex x=4,~~y=3$
$latex \begin{cases}-2x-y=1 \\ 3x+4y=6 \end{cases}$, solution: $latex x=-2, ~~y=3$
$latex \begin{cases}3x+4y-27=0 \\ 5x+y-11=0 \end{cases}$, solution: $latex x=1,~~y=6$
Systems of equations with no solution
Systems of linear equations do not have a solution when the graphs of their equations are parallel. If the lines from the equations are parallel, the lines will never intersect.
In these cases, the equations of the system do not have any point in common, so there is no solution that can satisfy all the equations of the system. These types of systems of equations are inconsistent.
In the following graph, we can see a system of two equations with two unknowns that have no solution:
We can see that the graphs of the equations are parallel and do not intersect, so they do not have any points in common.
Therefore, when we have systems of linear equations, we can draw simple graphs of the equations to determine if the system has no solutions. If at least two of the lines are parallel, the system has no solution.
Algebraically, we can determine if a system of linear equations has no solution by comparing their slopes. If the equations have the same slope, the lines will be parallel.
EXAMPLES
Examples of systems of equations with no solution:
- $latex \begin{cases}x+y=10 \\ x+y=5 \end{cases}$
By writing the equations in the form $latex y=mx+b$, we know that the slope is m. In this case, we have the equations $latex y=-x+10$ and $latex y=-x+5$.
The slopes of both equations are equal to -1. Then, the lines are parallel and the system of equations has no solution.
- $latex \begin{cases}-2x+y=1 \\ -4x+2y=6 \end{cases}$
Writing the equations in the form $latex y=mx+b$, we have $latex y=2x+1$ and $latex y=2x+3$. This means that their slopes are equal to 2, so the lines are parallel and the system has no solution.
Systems of equations with infinite solutions
A system of linear equations has infinitely many solutions when the graphs of the equations are superimposed on each other. This happens when we have equivalent versions of the same equation.
In this case, we will have two or more equivalent linear equations, so any solution that works for the first equation will also work for the second.
Systems of equations that have an infinite number of solutions are consistent since they have at least one solution.
In the following graph, we can observe a system of two equations with two unknowns that has an infinite number of solutions:
We can see that the lines are superimposed. Basically, this means that the given equations result in the same line when graphed.
Therefore, to determine if a system of equations has infinite solutions, we can draw basic graphs of the equations and see if the equations overlap.
Algebraically, we can determine if the equations of the system are equivalent by simplifying and manipulating them. For example, we might have one of the equations equal the other when multiplied by 2.
EXAMPLES
Examples of systems of equations with infinite solutions:
- $latex \begin{cases}x+y=3\\ 2x+2y=6 \end{cases}$
The equations of the system appear to be different at first glance. However, the second equation is obtained by multiplying the first by 2. Therefore, these equations are equivalent, and the system will have infinite solutions.
- $latex \begin{cases}-2x+y=1 \\ 6x-3y=-3 \end{cases}$
The second equation is obtained by multiplying the first by -3. This means that the equations are equivalent, and the system will have infinite solutions.
See also
Interested in learning more about systems of equations? Take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
