Question

OKay so already solved for a with @vocaloid now for part b and c

Answer 1

@Vocaloid

Answer 2

first we take the coefficients and put them into a matrix

Answer 3

make sure to keep all signs (+ or -)

Answer 4

\[\left[\begin{matrix}3 & 2 \\ 2 & -4\end{matrix}\right]\]

Answer 5

\[\left(\begin{matrix}x \\ y\end{matrix}\right)\]

Answer 6

okay so would that be 14,4

Answer 7

good, so \[\left(\begin{matrix}14 \\ ?\end{matrix}\right)\]

Answer 8

4

Answer 9

should be 14,4

Answer 10

good, so now we write these in order

Answer 11

now, in order to isolate the x,y matrix we multiply both sides by the inverse of the coefficient matrix

Answer 12

need to have the x, y, matrix too

Answer 13

^it should look like what I drew up there

Answer 14

good, now we find the inverse of the coefficient matrix (the 3,2,2,-4)

Answer 15

okay

Answer 16

so we subtract 1 from each value

Answer 17

\(\color{#0cbb34}{\text{Originally Posted by}}\) @Vocaloid
\[\left[\begin{matrix}3 & 2 \\ 2 & -4\end{matrix}\right]\]
\(\color{#0cbb34}{\text{End of Quote}}\)

apply this logic to this matrix to find the determinant

Answer 18

2,1,1,-5

Answer 19

a = 3, b = 2, c = 2, d = -4

Answer 20

the -1 in the top right corner does not mean -1, this is the notation that means inverse

Answer 21

-16

Answer 22

good, the determinant is 16, now we find this part:

Answer 23

switch the a and d values
then multiply the c and b values by -1 to find this new matrix

Answer 24

a = -4, b = 2, c = 2, d = 3

Answer 25

like that right?

Answer 26

I switched the values and now I apply the formula

Answer 27

?

Answer 28

almost, we also need to multiply b and c by -1

Answer 29

-6

Answer 30

\[\left[\begin{matrix}-4 & -2 \\ -2 & 3\end{matrix}\right]\]

Answer 31

6

Answer 32

our work so far should be a 2x2 matrix

Answer 33

so now we apply the inverse formula by taking this new matrix and multiplying it by 1/the determinant which is 1/(-16)

Answer 34

\(\color{#0cbb34}{\text{Originally Posted by}}\) @Vocaloid
\[\frac{ 1 }{ -16 }\[\left[\begin{matrix}-4 & -2 \\ -2 & 3\end{matrix}\right]\]\]
\(\color{#0cbb34}{\text{End of Quote}}\)

Answer 35

then we left-multiply this matrix * each side of the original equation

Answer 36

the left side cancels out (thankfully)

Answer 37

when you multiply the black parts out you should get x = 4 and y = 1 which agrees with the solution we obtained by solving the system

Answer 38

[make sure you are applying the matrix multiplication method not just multiplying across the rows]

Answer 39

Yes I applied that and it checked to be the correct answer

Answer 40

awesome, and then for part c you would just graph them and confirm that they intersect at (4,1)

Answer 41

SO just graph the two systems of equations given

Answer 42

yes

Answer 43

H

Answer 44

good, now be sure to mark the intersection as (4,1)

Answer 45

just make a point at (4,1) and write (4,1) besides it

Answer 46

awesome, that's it for c