EvLer said:is every system of 6 equations and 5 unknowns consistent?
I think that not every one is consistent because after Gaussian elimination you may arrive at equation where 0=1, but I want a sure confirmation.
A system of linear equations is a set of two or more equations that involve one or more variables, where each equation is linear (meaning the highest power of the variable is 1). The solutions to a system of linear equations are the values of the variables that make all of the equations true simultaneously.
There are several methods for solving a system of linear equations, including graphing, substitution, and elimination. In graphing, the equations are graphed on the same coordinate plane and the solution is where the lines intersect. In substitution, one equation is solved for one variable and then substituted into the other equation to solve for the other variable. In elimination, the equations are manipulated to eliminate one variable, and then the remaining equation is solved for the remaining variable.
Yes, a system of linear equations can have one, infinite, or no solutions. If the equations are consistent (they have at least one solution), they can have one solution if the lines intersect at one point, infinite solutions if the lines are the same (they are equivalent), or no solutions if the lines are parallel and do not intersect.
A consistent system of linear equations has at least one solution, meaning the equations intersect at one point. An inconsistent system has no solutions, meaning the equations are parallel and do not intersect. This can also be thought of as having no common solution, or the solution set being empty.
Yes, a system of linear equations can have any number of variables. The number of equations must match the number of variables in order to have a unique solution, but there are infinitely many possible solutions for systems with more variables than equations. These types of systems are often solved using matrices and other advanced methods.