Systems of linear equations.

So what were linear equations?

linear differential equations with constant coefficients.

Let's solve this set of linear equations.

of linear equations.

I would say a lot easier than what we did in the previous first order homogeneous difference equations, or the exact equations.

Those are called homogeneous linear differential equations, but they mean something actually quite different.

This just seems like these linear equations are pretty easy to solve for.

Now I can go back from this world, back to my linear equations.

You have a system of linear equations which is Algebra I.

And this works every time for second order homogeneous constant coefficient linear equations.

And if you're doing this in you Algebra 1 or your Algebra 2 class you're probably using it to represent linear equations.

So linear equations are A x equal b and the idea now is to solve this

linear equations. and you might be saying well you know, this is an equation,

Is the system of linear equations below dependent or independent, and they gave us two equations right here.

In a mathematical programming model, if the objective functions and constraints are represented entirely by linear equations, then the model is regarded as a linear model.

So, just to start off with, if we're dealing with systems of linear equations in two dimensions, there's only three possibilities that the lines, or the equations, can have relative to each other.

But anyway, these are useful properties to maybe internalize for second order homogeneous linear differential equations.

But we're dealing with three dimensional linear equations, so this is just going to be a plane.

History The study of linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations.

let's see if this system of linear equations right here is dependent or independent.

One, trying to figure out whether the equations are exact.

Partial differential equations we'll get into later.

But actually the real place where it you really, really pays off is, let's say that you have a bunch of linear equations to solve.

So if you start to feel like this is something like what you learned when you learned solving systems of linear equations, that's no coincidence.

Please. One, one, please.

I figure it never hurts getting as much practice as possible solving systems of linear equations, so let's solve this one.

I'll come at once.

Right! Lower away together!

We're asked to convert these linear equations into slope intercept form and then graph them on a single coordinate plane.

If someone told you just the traditional way that you've done it, either through substitution, or through adding or subtracting the linear equations you wouldn't be able to find a solution where these two intersect, if a b is equal to c d.

Shukla Bose Teaching one child at a time

But we began with the idea of one child at a time.

In an equation, you typically have one solution, or at least the ones we've solved so far.

And like the first video, where I talked about reduced row echelon form, and solving systems of linear equations using augmented matrices, at least my gut feeling says,

All at once, and much... much too completely.

Lower away together!

Lower away together!

For example, factoring quadratics with leading coefficients greater than one.

If you don't remember how to do this you should probably review the level one and level two linear equations lectures that I have in the algebra playlist.

And I'm getting these problems from page 80 of my old college differential equations books.

Everything's possible simultaneously.

Now these are one way they're not cycles one way linear changes that had profound and amazingly predictable consequences.

We have one time ... To understand ...

I have separated the equations.