I have searched through many answers here already, some of them are similar but none of them talk in laymans terms. I would appreciate if someone who answers could tell me how to do these sorts of problems without all the fancy lingo.
x + 2y = 3
ax + by = -9
So I have no idea how to use a matrix to solve this. Our teacher hasn't really shown us how to do this type of stuff. He is the type of teacher who lectures for an hour with no examples and then throws you to the sharks, sink or swim.
All I know from searching google and using desmos to graph my proposed solutions that:
(a) Is a value that represents the two lines been parallel to each other which means a and b must be a=1 and b=2
(b) When the lines are the same, this is the one I cant figure out.
(c) Any number for a and b other than when a is equal to 1, because that makes them parallel to each other.
Can someone explain to me in laymans terms how to use a matrix to figure this out for all parts a through c.
Thank you.
Best Answer
Written out as a matrix equation, you have $$\begin{bmatrix}1&2\\a&b\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}3\\-9\end{bmatrix}$$ The determinant of the coefficient matrix is b-2a. Bad things happen if the determinant =0. If $ b \neq 2a$, multiply the first equation by b and the second equation by 2 and subtract. y has been eliminated,you can solve for x and substitute back to find x, so in this case there is a unique solution. If b=2a, the left side of the second equation is a times the first equation, so in the case that a=-3, b=-6, the second equation is -3 times the first equation and thus gives no new information and there are infinitely many solutions. If b=2a and $ a\neq -3$, there is a contradiction, i.e. there are no solutions.