It means that at least one of the equations can be expressed as a linear combination of some of the other equations.
A linear combination of equations is the addition (or subtraction) of equations. And since an equation can be added several times, it includes multiples of equations.
For example, if you have
x + 2y = 3
and
2x + y = 4
Then adding 2 times the first and 3 times the second gives
8x + 7y = 18
This is, therefore, dependent on the other 2.
If you have n unknown variables, there will be a unique solution if, and only if, you must have a set of n independent linear equations.
No....not necessary
if a dependent system of equation is solved, how many solutions will there be?
Yes
When its matrix is non-singular.
dependent
It is a system of linear equations which does not have a solution.
The terms consistent and dependent are two ways to describe a system of linear equations. A system of linear equations is dependent if you can algebraically derive one of the equations from one or more of the other equations. A system of linear equations is consistent if they have a common solution.An example of a dependent system of linear equations:2x + 4y = 84x + 8y = 16Solve the first equation for x:x = 4 - 2yPlug that value of x into the second equation:16 - 8y + 8y = 16, which gives 16 = 16.No new information was gained from the second equation, because we already knew 16 = 16, so these two equations are dependent.An example of an inconsistent system of linear equations:Because consistency is boring.2x + 4y = 84x + 8y = 15Solve the first equation for x:x = 4 - 2yPlug that value of x into the second equation:16 - 8y + 8y = 15, which gives 16 = 15.This is a contradiction, because 16 doesn't equal 15. Therefore this system has no solution and is inconsistent.
No....not necessary
One equation is simply a multiple of the other. Equivalently, the equations are linearly dependent; or the matrix of coefficients is singular.
if a dependent system of equation is solved, how many solutions will there be?
A dependent system is defined as "a system of equations that has infinite solutions." It is an equation that is used in various mathematical situations.
Yes
a linear equation
When its matrix is non-singular.
dependent
This is a linear algebra question and it is incomplete since there are no equation which have to be solved.
By elimination and substitution