Question

How to use Cramer’s rule to solve the following system of equations

5x+3y=7
4x + 5y =3

Answer 1

can you make the matrix of coefficients ?

Answer 2

no

Answer 3

write the numbers in front of the x and y
to make a row

Answer 4

\[\left[\begin{matrix}5 & 3 \\ 4 & 5\end{matrix}\right]\left[\begin{matrix}x \\ y \end{matrix}\right]=\left[\begin{matrix}7 \\ 3\end{matrix}\right]\]

Answer 5

do you know how to find the determinant of a 2 x 2 matrix ?

Answer 6

no all this is new for me

Answer 7

for a 2 x 2 matix
A B
C D

the determinant is AD - BC

can you find the determinant of the matrix ?

Answer 8

no

Answer 9

sshayer wrote down the 2 x 2 matrix
do you see it ?

Answer 10

when you have time, see
https://www.khanacademy.org/math/precalculus/precalc-matrices/determinant-of-2x2-matrix/v/finding-the-determinant-of-a-2x2-matrix

Answer 11

\[or~AX=B\]
\[where ~A=\left[\begin{matrix}5 & 3 \\ 4 & 5\end{matrix}\right]\]
\[\left| A \right|=\left|\begin{matrix}5 & 3 \\ 4 & 5\end{matrix}\right|=5*5-4*3=25-12=13\]
\[\left| A _{1} \right|=\left|\begin{matrix}7 & 3 \\ 3 & 5\end{matrix}\right|=7*5-3*3=35-9=26\]
\[\left| A _{2} \right|=\left[\begin{matrix}5 & 7 \\ 4 & 3\end{matrix}\right]=5*3-4*7=15-28=-13\]
\[x=\frac{ \left| A _{1} \right| }{ \left| A \right| }=\frac{ 26 }{ 13 }=2\]
\[y=\frac{ \left| A _{2} \right| }{ \left| A \right| }=\frac{ -13 }{ 13 }=-1\]