Question

what is the augmented matrix

y - 3 = 2x
3x = -y

Answer 1

I'll show you how to do a similar example. Let's say we had this system

y - 5 = 9x
4x = -2y

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The goal is to get the two equations into `Ax+By = C` form. To do that for the first equation, we subtract 9x from both sides and add 5 to both sides

y - 5 = 9x
y - 5-9x = 9x-9x
y - 5-9x = 0
y - 5-9x+5 = 0+5
-9x+y = 5


Now onto the second equation 4x = -2y. Let's add 2y to both sides to get

4x = -2y
4x+2y = -2y+2y
4x+2y = 0


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So we have this new equivalent system

-9x+y = 5
4x+2y = 0


which is the same as this system


-9x+1y = 5
4x+2y = 0

The system above is in the form

Ax+By = C
Dx+Ey = F

where...

A = -9
B = 1
C = 5
D = 4
E = 2
F = 0

Now just plug all this into the augmented matrix below

\[\Large \left[\begin{array}{cc|c}A & B & C\\D & E & F\end{array}\right]\]

to get

\[\Large \left[\begin{array}{cc|c}-9 & 1 & 5\\4 & 2 & 0\end{array}\right]\]

and that's all there is to it. Hopefully this example is enough to get you pointed in the right direction to answer your question.