Concepts:

Least Square Method:

It is an approximation method to minimize the error.

In this method, the distance between the estimated value and the actual value is minimized.

According to the least square method, \({\rm{R\;}} = \mathop \sum \limits_1^{\rm{n}} {\left( {\overline {{{\rm{x}}_{\rm{i}}}} - {\rm{x}}} \right)^2}\) where \(\overline {{{\rm{x}}_{\rm{i}}}}\) are the estimated (or guessed value) and x is the actual values.

By minimizing this distance R the least-squares error can be found out.

Calculations:

Given the functions are:\(2{\rm{x\;}} = {\rm{\;}}3 \Rightarrow 2{\rm{x}} - 3 = 0{\rm{\;and\;}}4{\rm{x}} = 1{\rm{\;}} \Rightarrow 4{\rm{x}} - 1 = 0{\rm{\;}}\)

∴ \({\rm{R}} = {\rm{\;}}{\left( {2{\rm{x}} - 3} \right)^2} + {\left( {4{\rm{x}} - 1} \right)^2}.{\rm{\;}}\)

Hence, to minimize the value of  \({\rm{R}},{\rm{\;\;}}\frac{{{\rm{dR}}}}{{{\rm{dx}}}} = 0\)

∴ \({\rm{\;}}\frac{{{\rm{dR}}}}{{{\rm{dx}}}} = 2 \times 2\left( {2{\rm{x}} - 3} \right) + 4 \times 2\left( {4{\rm{x}} - 1} \right) = 0{\rm{\;}}\)

∴ \({\rm{x}} = \frac{1}{2}{\rm{\;}}\& {\rm{\;}}{{\rm{R}}_{{\rm{min}}}} = {\left( {2 \times \frac{1}{2} - 3} \right)^2} + {\left( {4 \times \frac{1}{2} - 1} \right)^2} = 5{\rm{\;\;\;}}\)

∴ The value of x that minimizes the sum of squares of the errors in the two equations is 1/2.