Question

Solve the system of equations using either the substitution method or the multiplication/addition method
answer:
3x+2y=14
2x-4y=4

6x+4y=28
2x-4y=4
8x = 32
x = 4
2y = 14 - 12 = 2
y = 1
(x , y) = (4 , 1)
question:
Check your solution by writing the system as a matrix equation and using the inverse matrix.
(how do I do that?)

Answer 1

Your matrix would be made up of the coefficients in your equation:\[\left[\begin{matrix}3 & 2 \\ 2 & -4\end{matrix}\right] = \left(\begin{matrix}14 \\ 4\end{matrix}\right)\] The left side can be called your "A" matrix. Here is a link on how to do an inverse: http://mathworld.wolfram.com/MatrixInverse.html

Answer 2

oh ok so thats how a matrix is

Answer 3

You can also think of it like this:\[\left[\begin{matrix}3 & 2 \\ 2 & -4\end{matrix}\right] \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}14 \\ 4\end{matrix}\right)\]
So the first line would mean 3 for the x and 2 for the y, so 3x+2y=14, and the second line would mean 2 for the x and -4 for the y, so 2x-4y=4, so you can check that you wrote it in matrix form correctly.

Answer 4

so my work was correct since it works in the matrix?

Answer 5

Oh no, your work hasn't been checked yet. I just meant the matrix was written correctly because it comes out to be the same system of equations. I haven't mentioned solutions yet. That's what I gave you the link for -- it tells you step by step how to do an inverse. That's what you have to do first.

Answer 6

oh ok thank u!

Answer 7

To make it more clear, here is what it means when it asks you to check the solutions using the inverse: Your original matrix equation is in the form "Ax=b" So the "A" is the A matrix I said, the "x" is the (x,y) column next to it, and the "b" is the (14,4) on the right side. To find solutions, multiply both sides by the inverse:
\[Ax=b\]\[A ^{-1} (Ax)=A ^{-1} b\] But A^(-1) * A is just equal to the identity matrix (or just think of it as = 1), so: \[Ix=A ^{-1} b\] \[x=A ^{-1} b\] So if you find the inverse, you've found A^-1. Then just multiply A^-1 by b, and you already know what b is because Ax=b.. it's just the (14,4) column matrix. That gives you your solutions.

Answer 8

I can go through it step by step with you if you want after you've tried it yourself if you still are confused. Matrices are usually pretty easy, it just takes a ton of practice to get used to them, that's really all.

Answer 9

oh ok thank u for all your help! ill try an if i get stuck ill ask u