I am yet to read the answer to the previous function related question that I had asked. Thanks for the response.

I have a few more questions.

2 planes intersect at a line. Why is it called infinite solutions? I understand that algebraically two unknowns are expressed using the free variable z. y= 1-2z and x = 2-5z for example. But graphically, we are getting a line.

Similar to the case where 3 planes intersect at a line. However, here we say the system is consistent and has a solution. I am able to get it algebraically and not reconcile myself geometrically.

In the form y-y1 = m (x-x1) What is the intuition behind getting this form of the line. I mean is there a proof? How do I know that I have to write y-y1 and x-x1 ?

In a parabola when it has complex roots, we draw a parabola above X axis. How is the representation of this parabola on the Argand plane AND the complex 4D plane??? I do not want to get into complex analysis but if I can visualise x^2+1=0 in the complex world, I would be happy.

All of us accept that x+2y=4 is the same line as 2x+4y=8, both algebraically and graphically. I am learning row reduction. Multiplying an equation by a scalar leads to the same line.
I can say that if I take a point x1, I get the same y1 for both lines.

What is the meaning behind elimination? or subtracting 2 rows of a matrix when the coefficients are the same.
5x+ 10y = 20 and 5x+y=8, If I subtract both, I get that it means that 5x or x is the same in both lines so we equate it?

I get that x^2=-1 means x=± i.
The real part is 2D
The imaginary part is 2D

So, any complex number is 4D and saw this visualisation instead of the coloured ones which I do not understand. Redirect Notice

However, what I do not get is why is the real part 2D and the imaginary part 2D? Why is that so? A real number 3 say is considered 2D? And i is considered 2D? I do not get that.

Thanks for your continuing great questions . . . and, unfortunately due to our quite limited resources here, we’re going to have to restrict ourselves to answering questions that are Calculus related. As much as we’d love to help, material involving linear algebra and the “complex world” are beyond our current bandwidth to address.

I will answer your Question #2 here, though:

In the form y-y1 = m (x-x1) What is the intuition behind getting this form of the line. I mean is there a proof? How do I know that I have to write y-y1 and x-x1 ?

I imagine if I rewrite is as

m = \frac{y - y_1}{x - x_1}

you’ll recognize the line’s slope. If I gave you two points on a line \left(x_1, y_1\right) and \left(x_2, y_2\right), you’d use that equation to calculate the slope m . . . but if instead I give you the slope m and a point on the line \left(x_1, y_1\right), then you can see how the equation must hold true for any other point \left(x, y\right) on the line. Indeed, the equation you wrote is called the “Point-Slope Form” of a line for just that reason. And you’ll probably end up using this form a line a lot more frequently than the Slope-Intercept form (y = mx + b), including if you have to write the equation of the tangent or normal line to a function’s curve.

I’m sorry we can’t take the time to address your other questions, but trust you can find resources elsewhere – and we remain happy to try to help with any Calculus questions that arise for you!