The optimal values of the parameters in the least-squares line $y\=a x\+b$ are given in a compact form by the expressions

 $a\=\frac{\mathrm{\Σ} x y-n \stackrel{\&conjugate0;}{x} \stackrel{\&conjugate0;}{y}}{\mathrm{\Σ} x^2-n \stackrel{\&conjugate0;}{x}{}^2 }$    and    $b\=\frac{\stackrel{\&conjugate0;}{y} \mathrm{\Sigma } x^2- \stackrel{\&conjugate0;}{x} \mathrm{\Sigma } x y}{\mathrm{\Sigma } x^2-n {\stackrel{\&conjugate0;}{x}}^{}{}^2}$

where $\stackrel{\&conjugate0;}{x}\=\frac{1}{n}\sum _{k\=1}^n{x}_k$, $\stackrel{\&conjugate0;}{y}\=\frac{1}{n} \sum _{k\=1}^n{y}_k$, $\mathrm{\Σ} x^2\=\sum _{k\=1}^n{x}_k^2$, $\mathrm{\Σ} x y\=\sum _{k\=1}^n{x}_k y_k$.

These results are obtained by differentiating $S\=\sum _{k\=1}^n{\left(a x_k\+b-y_k\right)}^2$ with respect to $a$ and $b$ to form the equations

$\frac{\partial S}{\partial a}\=\sum _{k\=1}^n\left(2a{x}_k^2\+2b{x}_k-2x_k{y}_k\right)\=0$ and $\frac{\partial S}{\partial b}\=2nb\+\sum _{k\=1}^n\left(2a{x}_k-2y_k\right)\=0$ 

which become

$\left[\begin{array}{cc}n \stackrel{\&conjugate0;}{x}& n\\ \mathrm{\Σ} x^2& n \stackrel{\&conjugate0;}{x}\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right]\=\left[\begin{array}{c}n \stackrel{\&conjugate0;}{y}\\ \mathrm{\Σ} x y\end{array}\right]$ 

An application of Cramer's rule then gives

$a\=\frac{\|\begin{array}{cc}n \stackrel{\&conjugate0;}{y}& n\\ \mathrm{\Σ} x y& n \stackrel{\&conjugate0;}{x}\end{array}\|}{\|\begin{array}{cc}n \stackrel{\&conjugate0;}{x}& n\\ \mathrm{\Σ} x^2& n \stackrel{\&conjugate0;}{x}\end{array}\|}$ = $\frac{n^2 \stackrel{\&conjugate0;}{x} \stackrel{\&conjugate0;}{y} - n \mathrm{\Σ} x y}{n^2 \stackrel{\&conjugate0;}{x}{}^2-n \mathrm{\Σ} x^2}\=\frac{\mathrm{\Sigma } x y-n \stackrel{\&conjugate0;}{x} \stackrel{\&conjugate0;}{y}}{\mathrm{\Sigma } x^2-n \stackrel{\&conjugate0;}{x}{}^2 }$ 

and

$b\=\frac{\|\begin{array}{cc}n \stackrel{\&conjugate0;}{x}& n \stackrel{\&conjugate0;}{y}\\ \mathrm{\Σ} x^2& \mathrm{\Σ} x y\end{array}\|}{\|\begin{array}{cc}n \stackrel{\&conjugate0;}{x}& n\\ \mathrm{\Sigma } x^2& n \stackrel{\&conjugate0;}{x}\end{array}\|}$ = $\frac{n \stackrel{\&conjugate0;}{x} \mathrm{\Σ} x y-n \stackrel{\&conjugate0;}{y} \mathrm{\Σ} x^2}{n^2 \stackrel{\&conjugate0;}{x}{}^2-n \mathrm{\Sigma } x^2}$ = $\frac{\stackrel{\&conjugate0;}{y} \mathrm{\Sigma } x^2- \stackrel{\&conjugate0;}{x} \mathrm{\Sigma } x y}{\mathrm{\Sigma } x^2-n {\stackrel{\&conjugate0;}{x}}^{}{}^2}$