Question

Write the augmented matrix, and use row reduction to solve.
{2y=x+1
{3x-2=y

Possible answers:
A. (1, 1)
[-1 2 : 1]
[3 -1 : 2]
B. (1, 2)
[-1 2 : 1]
[3 -1 : 2]
C. (2, -1)
[-1 1 : 2]
[3 2 : -1]
D. (-1, 1)
[-1 1 : 2]
[3 2 : -1

Answer 1

ax+by = e
cx+dy = f

[ a b ] [ x ] = [ e ]
[ c d ] [ y ] = [ f ]


[ a b : e ]
[ c d : f ]

Answer 2

Is that how I will solve all equations like this one?

Answer 3

Yeah,
first get them into the form

ax + by = e
cx + dy = f

Answer 4

So would I subtract x from the first one to get -x+2y=1 and how would I switch around the second one?

Answer 5

{2y = x+1
{3x-2 = y


that's right for the first equation.
2y = x+1
(taking away x form both sides)
-x+2y=1

now for the second
3x-2 = y
(take away y from both sides)
(and add 2 to both sides)

Answer 6

So I would get 3x-y=2?

Answer 7

Good! you now have

{-x +2y = 1
{3x -y = 2

Answer 8

Which is of the form

ax+by = e
cx+dy = f


We can express this system of equation as a matrix equation

[ a b ] [ x ] = [ e ]
[ c d ] [ y ] = [ f ]

Answer 9

So my answer would be
[-1 2 : 1]
[3 -1 : 2]

Answer 10

Yeah, that is the right augmented system

Answer 11

but you still have to solve it for x and y

Answer 12

How do I do that?

Answer 13

you have to reduce the argument system so something like
[1 0 : g]
[0 1 : h]

which means the same thing as
[ 1 0 ] [ x ] = [ g ]
[ 0 1 ] [ y ] = [ h ]

ie x = g
and y = h

Answer 14

to reduce the augmented system
you can take steps of either
* multiplying entire rows by a constant
* adding rows together

Answer 15

[-1 2 : 1]
[3 -1 : 2]

first step might be to change the first row by multiplying by -1

Answer 16

like this:
R1 --> -R1

[1 -2 : -1]
[3 -1 : 2]

(now the element on the top left is a 1)

Answer 17

So my answer for x and y would be (1, 1)?

Answer 18

next step might be to change the second row
by taking away three times the first row

R2 --> R2 - 3R1

Answer 19

What do you get after this operation?

Answer 20

[0 -7 : -1]?

Answer 21

not quite
[ 1, -2 : -1]
[3-3(1), -1-3(-2) : 2-3(-1)]

[ 1, -2 : -1]
[3-3, -1+6 : 2+3]

[ 1, -2 : -1]
[ 0, -1+6 : 2+3]

Answer 22

[0 5 : 5]?

Answer 23

yeah we have
[ 1, -2 : -1 ]
[ 0, 5 : 5 ]
(the bottom left element is now 0)

now we should dived the second row by 5
R2 --> R2/5

Answer 24

[ 1, -2 : -1 ]
[ 0, 5/5 : 5/5 ]

Answer 25

[0 1 : 1]

Answer 26

yeah we have

[ 1, -2 : -1 ]
[ 0, 1 : 1 ]

(only one element left to go)

Now, add two times the second row , to the first row
R1 --> R1 + 2R2

Answer 27

[1 0 : 1]?

Answer 28

Great!

We have
[ 1, 0 : 1 ]
[ 0, 1 : 1 ]

aka
[ 1 0 ] [ x ] = [ 1 ]
[ 0 1 ] [ y ] = [ 1 ]

which means
1x+0y = 1
0x+1y = 1

ie
x = 1
y = 1

Answer 29

You might like to check that this solution solve both of the original equations.

Answer 30

Thank you so much for your help!