Question

Using Cramer’s Rule, what is the minimum number of determinants that are needed to solve for all unknowns in the system of linear equations below?

10x-y=3
5x-2y=-24

Answer 1

Solve the system of equations

`\left\{\begin{array}{l}10x-y=3\\5x-2y=-24\end{array}\right.`

using Cramer’s Rule.

1. Find the determinants:

`\Delta=\left|\begin{array}{cc}10 & -1\\5 & -2\end{array}\right|=10\cdot (-2)-(-1)\cdot 5=-20+5=-15.`

`\Delta_x=\left|\begin{array}{cc}3 & -1\\-24 & -2\end{array}\right|=3\cdot (-2)-(-1)\cdot (-24)=-6-24=-30.`

`\Delta_y=\left|\begin{array}{cc}10 & 3\\5 & -24\end{array}\right|=10\cdot (-24)-3\cdot 5=-240-15=-255.`

2. Now find unknown variables:

`x=(\Delta_x)/(\Delta)=(-30)/(-15)=2,\\ \\y=(\Delta_y)/(\Delta)=(-255)/(-15)=17.`

Answer: the minimum number of determinants that are needed to solve for all unknowns in the system of linear equations is 3.