The set of equations, $x_1 + 3x_2 = 17$, $5x_1 + 7x_2 = 19, and \11x_1 + 13x_2 = 23$has no real solution, but the pseudoinverse may be used to find close values. First, the matrix$A$is set up:$A = \begin{bmatrix} 1 & 3 5 & 7 11 & 13 \end{bmatrix}$. We solve$A^\dagger = (A^TA)^{-1}A^T$and get$A^\dagger = \begin{bmatrix} -0.5197 & -0.2171 & 0.2368 0.4276 & 0.2039 & -0.1316 \end{bmatrix}$. Next, we find$\vec{x} = A^\dagger\vec{y}$and multiply:$\begin{bmatrix} -0.5197 & -0.2171 & 0.2368 0.4276 & 0.2039 & -0.1316 \end{bmatrix}$$\begin{bmatrix} 17 19 23 \end{bmatrix} =$$\begin{bmatrix} -7.51 8.12 \end{bmatrix}$This means$x_1 \approx -7.51$and$x_2 \approx 8.12$.