Question

The sum of two numbers is 6. If we multiply first by 3 and add second number to it, we get 10. Form equations and solve by matrix method and ECT.

Answer 1

Let the two numbers be x and y
x + y = 6
3x + y = 10
Writing these equations in matrix form
A x = B
$\left[\begin{array}{cc}1& 1\\ 3& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}6\\ 10\end{array}\right]$
$X=A^{-1}B$
$\left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{1-3}\left[\begin{array}{cc}1& -1\\ -3& 1\end{array}\right]\left[\begin{array}{c}6\\ 10\end{array}\right]=\left[\begin{array}{cc}-\frac{1}{2}\frac{1}{2}& \\ \frac{3}{2}& -\frac{1}{2}\end{array}\right]\left[\begin{array}{c}6\\ 10\end{array}\right]=\left[\begin{array}{c}6\times -\frac{1}{2}+10\times \frac{1}{2}\\ 6\times \frac{3}{2}-10\times \frac{1}{2}\end{array}\right]=\left[\begin{array}{c}2\\ 4\end{array}\right]\Rightarrow x=2y=4$