# A system of linear equations with more equations than unknowns is sometimes called an overdetermined system. Can such a system be consistent? Illustrate your answer with a specific system of three equations in two unknowns.

A system of linear equations with more equations than unknowns is sometimes called an overdetermined system. Can such a system be consistent? Illustrate your answer with a specific system of three equations in two unknowns.
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Nathalie Redfern

The system of linear equation with unknown lesser than the equations.
Step 2
Concept:
Over-determined system of equations is the system of linear equation with unknowns lesser than the equations.
Step 3
The over-determined system is inconsistent with the irregular coefficients. With linear equations they are consistent.
To consider a system with three equations but two unknowns,
${x}_{1}+{x}_{2}=3$
$2{x}_{1}-{x}_{2}=0$
$3{x}_{1}+3{x}_{2}=9$
The augmented matrix of the above equations is given below
$\left[\begin{array}{ccc}1& 1& 3\\ 2& -1& 0\\ 3& 3& 9\end{array}\right]$
To apply row application
$\left[\begin{array}{ccc}1& 1& 3\\ 2& -1& 0\\ 3& 3& 9\end{array}\right]\sim \left[\begin{array}{ccc}1& 1& 3\\ 0& -3& -6\\ 0& 0& 0\end{array}\right]$
To simplify row 1 and 2
$\left[\begin{array}{ccc}1& 1& 3\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]\sim \left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]$
The corresponding system of equation is given below,
${x}_{1}=1$
${x}_{2}=2$
The system of equation is consistent and have its unique solution

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Jeffrey Jordon

Step 1

If we have m equations and n variables where m>n (more equations than variables), then system can be consistent if last m-n equations are linear combinations of previous ones.

For example:

x+y=1

x-y=1

3x+y=3

You can see that third equation is

Solution of system is (x,y)=(1,0)

Result:

If we have m equation and n variables where m>n (more equations than variables), then system can be consistent if last m-n equations are linear combinations of previous ones.

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