Answer:
(a) x=2, y=-1
(b) x=2, y=2
(c) [tex]\displaystyle x=\frac{5}{2}, y=\frac{5}{4}[/tex]
(d) x=-2, y=-7
Step-by-step explanation:
Cramer's Rule
It's a predetermined sequence of steps to solve a system of equations. It's a preferred technique to be implemented in automatic digital solutions because it's easy to structure and generalize.
It uses the concept of determinants, as explained below. Suppose we have a 2x2 system of equations like:
[tex]\displaystyle \left \{ {{ax+by=p} \atop {cx+dy=q}} \right.[/tex]
We call the determinant of the system
[tex]\Delta=\begin{vmatrix}a &b \\c &d \end{vmatrix}[/tex]
We also define:
[tex]\Delta_x=\begin{vmatrix}p &b \\q &d \end{vmatrix}[/tex]
And
[tex]\Delta_y=\begin{vmatrix}a &p \\c &q \end{vmatrix}[/tex]
The solution for x and y is
[tex]\displaystyle x=\frac{\Delta_x}{\Delta}[/tex]
[tex]\displaystyle y=\frac{\Delta_y}{\Delta}[/tex]
(a) The system to solve is
[tex]\displaystyle \left \{ {{x+y=1} \atop {x-2y=4}} \right.[/tex]
Calculating:
[tex]\Delta=\begin{vmatrix}1 &1 \\1 &-2 \end{vmatrix}=-2-1=-3[/tex]
[tex]\Delta_x=\begin{vmatrix}1 &1 \\4 &-2 \end{vmatrix}=-2-4=-6[/tex]
[tex]\Delta_y=\begin{vmatrix}1 &1 \\1 &4 \end{vmatrix}=4-3=3[/tex]
[tex]\displaystyle x=\frac{\Delta_x}{\Delta}=\frac{-6}{-3}=2[/tex]
[tex]\displaystyle y=\frac{\Delta_y}{\Delta}=\frac{3}{-3}=-1[/tex]
The solution is x=2, y=-1
(b) The system to solve is
[tex]\displaystyle \left \{ {{4x-y=6} \atop {x-y=0}} \right.[/tex]
Calculating:
[tex]\Delta=\begin{vmatrix}4 &-1 \\1 &-1 \end{vmatrix}=-4+1=-3[/tex]
[tex]\Delta_x=\begin{vmatrix}6 &-1 \\0 &-1 \end{vmatrix}=-6-0=-6[/tex]
[tex]\Delta_y=\begin{vmatrix}4 &6 \\1 &0 \end{vmatrix}=0-6=-6[/tex]
[tex]\displaystyle x=\frac{\Delta_x}{\Delta}=\frac{-6}{-3}=2[/tex]
[tex]\displaystyle y=\frac{\Delta_y}{\Delta}=\frac{-6}{-3}=2[/tex]
The solution is x=2, y=2
(c) The system to solve is
[tex]\displaystyle \left \{ {{-x+2y=0} \atop {x+2y=5}} \right.[/tex]
Calculating:
[tex]\Delta=\begin{vmatrix}-1 &2 \\1 &2 \end{vmatrix}=-2-2=-4[/tex]
[tex]\Delta_x=\begin{vmatrix}0 &2 \\5 &2 \end{vmatrix}=0-10=-10[/tex]
[tex]\Delta_y=\begin{vmatrix}-1 &0 \\1 &5 \end{vmatrix}=-5-0=-5[/tex]
[tex]\displaystyle x=\frac{\Delta_x}{\Delta}=\frac{-10}{-4}=\frac{5}{2}[/tex]
[tex]\displaystyle y=\frac{\Delta_y}{\Delta}=\frac{-5}{-4}=\frac{5}{4}[/tex]
The solution is
[tex]\displaystyle x=\frac{5}{2}, y=\frac{5}{4}[/tex]
(d) The system to solve is
[tex]\displaystyle \left \{ {{6x-y=-5} \atop {4x-2y=6}} \right.[/tex]
Calculating:
[tex]\Delta=\begin{vmatrix}6 &-1 \\4 &-2 \end{vmatrix}=-12+4=-8[/tex]
[tex]\Delta_x=\begin{vmatrix}-5 &-1 \\6 &-2 \end{vmatrix}=10+6=16[/tex]
[tex]\Delta_y=\begin{vmatrix}6 &-5 \\4 &6 \end{vmatrix}=36+20=56[/tex]
[tex]\displaystyle x=\frac{\Delta_x}{\Delta}=\frac{16}{-8}=-2[/tex]
[tex]\displaystyle y=\frac{\Delta_y}{\Delta}=\frac{56}{-8}=-7[/tex]
The solution is x=-2, y=-7
