Questions: Solve the system x₁ x₂ x₃ x₄ x₅ x₆ = square square square square + s square square square + t square square square square + u square square square square square square square

Solution Steps

To solve the given system, we need to express the vector \([x_1, x_2, x_3, x_4, x_5, x_6]\) as a linear combination of the given vectors and parameters \(s\), \(t\), and \(u\). This involves setting up a system of linear equations based on the components of the vectors and solving for the unknowns \(s\), \(t\), and \(u\).

We are given a system of equations based on the linear combination of vectors. The equations are: \[ \begin{align_} s + 2t + 3u + 1 &= 10, \\ s + 2t + 3u + 2 &= 20, \\ s + 2t + 3u + 3 &= 30, \\ s + 2t + 3u + 4 &= 40, \\ s + 2t + 3u + 5 &= 50, \\ s + 2t + 3u + 6 &= 60. \end{align_} \]

Each equation can be simplified by subtracting the constant term on the right-hand side: \[ \begin{align_} s + 2t + 3u &= 9, \\ s + 2t + 3u &= 18, \\ s + 2t + 3u &= 27, \\ s + 2t + 3u &= 36, \\ s + 2t + 3u &= 45, \\ s + 2t + 3u &= 54. \end{align_} \]

The simplified equations are inconsistent because they suggest that the same expression \(s + 2t + 3u\) equals different values (9, 18, 27, 36, 45, 54). This indicates that there is no solution to the system of equations as it stands.

Final Answer