So we were able to solve this system of equations.

0.2 times x is equal to 4.

learned before, but it's a neat trick.

When we say system of equations, we just mean many equations that have many unknowns.

So, I'll see you in the part 2 of using the quadratic equation.

But anyway I don't want to get caught up in terminology.

solve the systems of linear equations by graphing. and they give us two equations here.

So the first thing we want to do here is not to factor this

The boy solved the simultaneous equation with ease.

You might have to use a quadratic equation or something.

Right?

There's really not much new here.

And we've factored our binomial.

And then the right hand side will be equal to 0.

And this is a good idea to memorize.

And maybe I'll show you in a future video, the quadratic equation and I think I've already done one where I proved the quadratic equation.

I will now introduce you to the idea of a homogeneous differential equation.

Bx plus C is equal to 0.

And we are done.

I just subtracted 16 from both sides to get here.

Let's say we have y is equal to 3x minus 6.

And let's say we try to do this, and it's not separable, and it's not exact.

leading coefficient.

All right, this is another one that should be cut and pasted. All right, four steps to derive the quadratic formula are shown below.

We first captured the dynamics with a set of differential equations.

And what we're dealing with are going to be first order equations.

Well, the highest derivative in it is just the first derivative, so the order is equal to 1.

So x 1 amp y 7 does not satisfy the second equation.

But I really want you to understand the graphical nature of solving systems of equations.

Let's say I wanted to solve minus 8x squared plus 5x plus 9.

Now we have a quadratic expression equaling 0.

So let's say I have a quadratic equation.

Let's do one more homogeneous differential equation, or first order homogeneous differential equation, to differentiate it from the homogeneous linear differential equations we'll do later.

We're going to start embarking on higher order differential equations.

Equation

Equation

Once again, whenever you see anything like this, get all of your terms on one side of the equation and get a 0 on the other side.

If it's not separable, but you can still put it in this form, you say, hey, is it an exact equation?

But with that said, any number times this imaginary unit i is an imaginary number.

All right, what are the solutions for the quadratic equation x squared plus 6x is equal to 16?

Divided by 2xy.

So this example I just wrote here, this is a second order

For deflections further out you'll see only distortions but generally not multi split images, that's called the weak lensing regime.

I think I'll eventually do a play list on essentially an introductory course on differential equations that you would take at college.

And this implicitly defined function, or curve, or however you want to call it, is the solution to our original homogeneous first order differential equation.