# Given a homogeneous system of linear equations, if the system is overdetermined, what are the possibilities as to the number of solutions?

Question
Equations
Given a homogeneous system of linear equations, if the system is overdetermined, what are the possibilities as to the number of solutions?

2021-01-03
Step 1
Given:
An overdetermined homogeneous system of linear equations.
Step 2
The homogeneous system of linear equations is said to be overdetermined if the number of equations m is more than the number of unknowns n.
If all the equations are linearly independent and m >n then there is no solution.
Step 3
If the system of equation is consistent, then, in this case, there is either one solution or set of solutions.
If all the equations are not linearly independent and s out of m equations are linearly independent, and if
(i) s>n, then there is no solution.
(ii) If s=n, then the system has either one solution or no solution.
(iii) If s For example:
Consider the homogeneous system of linear equations.
x+y=0, 2x+3y=0, 3x+2y=0
Here the number of equations is m=3 and the number of unknowns n=2.
As m>n, therefore, the system is overdetermined.
x=y=0 satisfy the equations.

### Relevant Questions

Given a homogeneous system of linear equations, if the system is overdetermined, what are the possibilities as to the number of solutions?
Explain.
Consider the following system of linear equations:
4x+2y=25
4x-y=-5
If the value of y is 10 what is the value of X for this system:
1.1.25
2.11.25
3.1.45
4.5
Use Cramer's rule to solve the given system of linear equations.
$$\displaystyle{x}_{{{1}}}-{x}_{{{2}}}+{4}{x}_{{{3}}}=-{2}$$
$$\displaystyle-{8}{x}_{{{1}}}+{3}{x}_{{{2}}}+{x}_{{{3}}}={0}$$
$$\displaystyle{2}{x}_{{{1}}}-{x}_{{{2}}}+{x}_{{{3}}}={6}$$
Use Cramer’s Rule to solve (if possible) the system of linear equations.
13x-6y=17
26x-12y=8
Solve the given system of linear equations.
$$\displaystyle{\frac{{{1}}}{{{2}}}}{x}-{\frac{{{3}}}{{{4}}}}{y}={0}$$
8x-12y=0
(x,y)=
Solve the system. If the system does not have one unique solution, also state whether the system is onconsistent or whether the equations are dependent.
2x-y+z=-3
x-3y=2
x+2y+z=-7
Find all the solutions of the system of equations:
x+2y-z=0, 2x+y+z=0, x-4y+5z=0.
$$\displaystyle{b}{e}{g}\in{\left\lbrace{c}{a}{s}{e}{s}\right\rbrace}{x}&-{y}&+{5}{z}&={26}\backslash&\ \ \ {y}&+{2}{z}&={1}\backslash&&\ \ \ \ \ {z}&={6}{e}{n}{d}{\left\lbrace{c}{a}{s}{e}{s}\right\rbrace}$$
$$\displaystyle{x}^{{{2}}}+{2.20}{x}+{1.21}={0}$$