91Ó°ÊÓ

Consider the linear equations with more equations than unknowns as shown below:

Let \({x_1}\) and \({x_2}\) be two unknowns.

\(\begin{array}{l}{a_1}{x_1} + {b_1}{x_2} = {c_1}\\{a_2}{x_1} + {b_2}{x_2} = {c_2}\\{a_3}{x_1} + {b_3}{x_2} = {c_3}\end{array}\)

Here, \({a_1}\), \({b_1}\), \({c_1}\), \({a_2}\), \({b_2}\), \({c_2}\), \({a_3}\), \({b_3}\), and \({c_3}\) are constants.

A system of three linear equations in two unknowns is consistent if the coefficients of the first variables of any two linear equations are similar, and the coefficients of the second variables are opposite in sign but equal in magnitude.

So, let \({a_1} = {a_2} = 1\), \({b_1} = 1\), \({b_2} = - 1\), \({a_3} = 4\), \({b_3} = 5\), \({c_1} = 4\), \({c_2} = 0\), and \({c_3} = 10\).

Substitute all the coefficients \({a_1} = {a_2} = 1\), \({b_1} = 1\), \({b_2} = - 1\), \({a_3} = 4\), \({b_3} = 5\), \({c_1} = 4\), \({c_2} = 0\), and \({c_3} = 10\) in the system of equations \(\left\{ \begin{array}{l}{a_1}{x_1} + {b_1}{x_2} = {c_1}\\{a_2}{x_1} + {b_2}{x_2} = {c_2}\\{a_3}{x_1} + {b_3}{x_2} = {c_3}\end{array} \right.\).

\(\begin{array}{c}{x_1} + {x_2} = 4\\{x_1} - {x_2} = 0\\4{x_1} + 5{x_2} = 10\end{array}\)

By solving the equations \({x_1} + {x_2} = 4\) and \({x_1} - {x_2} = 0\), the value of \({x_1}\) can be obtained.

Thereafter, the value of \({x_1}\) can be substituted in the equation \(4{x_1} + 5{x_2} = 10\) to get the value of \({x_2}\).

Thus, a system of linear equations with more equations than unknowns can be consistent.