Solve the system of equations
[tex]\left\{\begin{array}{l}10x-y=3\\5x-2y=-24\end{array}\right.[/tex]
using Cramer’s Rule.
1. Find the determinants:
[tex]\Delta=\left|\begin{array}{cc}10 & -1\\5 & -2\end{array}\right|=10\cdot (-2)-(-1)\cdot 5=-20+5=-15.[/tex]
[tex]\Delta_x=\left|\begin{array}{cc}3 & -1\\-24 & -2\end{array}\right|=3\cdot (-2)-(-1)\cdot (-24)=-6-24=-30.[/tex]
[tex]\Delta_y=\left|\begin{array}{cc}10 & 3\\5 & -24\end{array}\right|=10\cdot (-24)-3\cdot 5=-240-15=-255.[/tex]
2. Now find unknown variables:
[tex]x=\dfrac{\Delta_x}{\Delta}=\dfrac{-30}{-15}=2,\\ \\y=\dfrac{\Delta_y}{\Delta}=\dfrac{-255}{-15}=17.[/tex]
Answer: the minimum number of determinants that are needed to solve for all unknowns in the system of linear equations is 3.
