If you have more equations than variables, then you either have no solution or an infinite number of solutions.
And now, a closer look
When there are more equations than variables in a system, it is called an overdetermined system. The consequences of having such a system are significant, as it affects the solvability of the system.
One possible outcome is that the system may have no solution. This occurs when the equations are contradictory, meaning that they cannot all be true at the same time. For example, if we have three equations with two variables and they are all lines that never intersect, there is no point where all three lines meet, and we say the system has no solution.
Alternatively, an overdetermined system can also have an infinite number of solutions. This happens when the equations are not contradictory but are overly restrictive. In other words, it doesn’t specify the values of all the variables uniquely. In these cases, there will be a parameter or parameters that allow us to express the solution set in terms of a formula involving those parameters.
George Pólya, a famous mathematician, once said, “Problems worthy of attack prove their worth by hitting back.” The problem of an overdetermined system is undoubtedly worth addressing but can present challenges when finding its solution(s).
Here are some interesting facts regarding overdetermined systems:
- Overdetermined systems arise frequently in fields such as physics, engineering, and data fitting.
- Overdetermined systems can be solved using methods such as least-squares approximation and singular value decomposition.
- In some cases, it is possible to reduce an overdetermined system to an equivalent system with fewer equations or variables.
- An overdetermined system with no solution can indicate that there may be errors or inconsistencies in the data used to formulate the system.
- An overdetermined system with an infinite number of solutions can imply that there is not enough data to identify a unique solution.
Table: Potential Outcomes of Overdetermined Systems
| Outcome | Definition |
|---|---|
| No Solution | The equations in the system are contradictory and cannot be true at the same time. |
| Infinite Solutions | The equations in the system are overly restrictive, and there is not enough data to specify all the variables uniquely. |
| Unique Solution | The equations in the system are not contradictory, and there is enough data to specify all the variables uniquely. |
A video response to “What happens if you have more equations than variables?”
In this video, the concept of underdetermined systems, where there are more variables than equations, is discussed using examples and augmented matrices with reduction steps. It is demonstrated that while some underdetermined systems have infinitely many solutions, others may have no solutions at all. The speaker provides examples of both scenarios to help viewers understand that the number of equations and variables can impact the solvability and solutions of a system.
See more answers
If we have more equations than variables, then the system will, in general, have no solution (unless some of the equations are linearly dependent). Such a system is said to be overdetermined or inconsistent .
In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. [citation needed] An overdetermined system is almost always inconsistent (it has no solution) when constructed with random coefficients.
If you have more equations than variables, then the system is called overdetermined. This system can have zero, one, or infinitely many solutions.
Such a system can have zero, one or infinitely many solutions.
Examples:
[math]\begin{cases} x=0 \ x=1 \end{cases}[/math]
has two equations and one variable, but no solution.
[math]\begin{cases} x=0 \ x=0 \end{cases}[/math]
has two equations and one variable, but one solution.
[math]\begin{cases} x+y=0 \ x+y=0 \ x+y=0 \end{cases}[/math]
has three equations and two variables, but infinitely many solutions.
One might concoct less obvious systems, but these are as good.
To the contrary, a system with less equations than variables either is inconsistent or it has infinitely many solutions; it’s impossible it has a unique solution.
Moreover, people are interested
Keeping this in consideration, How do you solve a system of equations with more than two variables? Response to this: SOLVE A SYSTEM OF EQUATIONS BY ELIMINATION.
- Write both equations in standard form.
- Make the coefficients of one variable opposites.
- Add the equations resulting from Step 2 to eliminate one variable.
- Solve for the remaining variable.
- Substitute the solution from Step 4 into one of the original equations.
Subsequently, How many solutions can there be to a homogeneous system with more equations than variables? infinitely many
If a homogeneous system of linear equations has more variables than equations, then it has a nontrivial solution (in fact, infinitely many). We will use elementary row operations to find the row-echlon matrix for the system.
One may also ask, Why a system with more variables than equations Cannot have a unique solution? The response is: If there are more variables than equations, you cannot find a unique solution, because there isnt one. However, you can eliminate some of the variables in terms of others. In other words, you can start the Gaussian elimination process and continue until you run out of rows.
Likewise, What is a system with fewer equations than variables? A system of polynomial equations which has fewer equations than unknowns is said to be underdetermined. It has either infinitely many complex solutions (or, more generally, solutions in an algebraically closed field) or is inconsistent.
Correspondingly, What happens if a system of equations is not solved? Answer will be: Those that are not solved for then form what is called a basis of the solution space, of your system of equations. The general solution of the system of equations is: assign arbitrary values to the basis variables: read off the others from the equations for them.
What happens if a polynomial equation is overdetermined? The reply will be: In the case of the systems of polynomial equations, it may happen that an overdetermined system has a solution, but that no one equation is a consequence of the others and that, when removing any equation, the new system has more solutions. For example, has the single solution but each equation by itself has two solutions.
What happens if you remove variables from an equation? In terms of eliminating variables from equations you find that on substituting for x and y, z drops out of the equation as well. All this means of course that your original equations were redundant, and you really had more variables than equations.
Can you find a unique solution if there are more variables? Response to this: If there are more variables than equations, you cannot find a unique solution, because there isnt one. However, you can eliminate some of the variables in terms of others. In other words, you can start the Gaussian elimination process and continue until you run out of rows.
Also, What happens if a system of equations is not solved? Those that are not solved for then form what is called a basis of the solution space, of your system of equations. The general solution of the system of equations is: assign arbitrary values to the basis variables: read off the others from the equations for them.
Considering this, How to solve equations with more than one variable? Example Question #5 : Equations With More Than One Variable Solve for X and Y for in the following pair of equations Possible Answers: Correct answer: Explanation: There are 2 ways to solve this set of equations. First, you can solve for one of the variables, then substitute that value for the variable in the other equation.
Subsequently, What happens when there are more linear equations than unknowns?
As a response to this: When there are more linear equations than unknowns, it is usually impossible to find a solution which satisfies all the equations. Then one often looks for a solution which approximately satisfies all the equations. Let a and c be known and x be unknown in the following set of equations where there are more equations than unknowns.
Keeping this in consideration, What happens if you remove variables from an equation? Response: In terms of eliminating variables from equations you find that on substituting for x and y, z drops out of the equation as well. All this means of course that your original equations were redundant, and you really had more variables than equations.