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## Algebra (all content)

### Unit 9: Lesson 11

Sal solves a system of a quadratic equation and a linear equation by graphing both equations and looking for their intersections, and then checks the solution algebraically. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• What is the difference between a linear or non-linear equation? •  Notice in the video that the equation where x was multiplied by a constant (-2), the graph was a straight line. The line was sloped, but it was straight. It was "linear". The equation that had x^2 made a curve, not a straight line.
• couldn't -x^2 be interpreted as -(x^2) or (-x^2)? It seems to me like it could be either one, and which one you should use would depend on how you arrived at the equation y = -x^2 + 6. If, for instance, you somehow came up with the variable -x, and then you realized that to get your answer you need to square it, this would be the case of (-x^2) which would result in a positive number. Or if you had your variable x, and you needed to square it and take the negative of it, you would be doing -(x^2). I'm sorry I can't think of any specific examples, but am I right in thinking that this can be interpreted in 2 ways, depending on the context? • How did you get the porabla at
I'm still confused.. • I think I understand your question. The line describing the parabola is curved, not straight, but with only a few points drawn in, it's hard to see why.

If you went through the equation and filled in "x" with all the points in between the points that he graphed, you would see a curve shape appear. When you only have two points it's easy to see why it looks like it should be straight. Sal just skipped over doing 2.5, and then 2.25, and so on. The more points you graph, the more it looks l
• Please explain the ^ symbol. • To find the points of intersection, you could also set the two equations equal to one another which gives you x=-2, +4 - plug each back into the easier of the equations and you get your related y-values. • If x^2 represents an upward parabola and -x^2 represents a downward parabola, what represents a sideways parabola? "sideways parabolas" are not functions and very seldom do you have to deal with any one of those.

This is becouse for any for every x there should be max 1 y value. So if you put in say 5 in the equation then you should not get two results. Two results would be the case if you have a sideways parabola.
• At to , I dont get how you can assume x=0 for the turning point • What you call the turning point we call a maximum or a minimum point, which as you observed is where the graph changes direction or turns.

The equation under consideration is y=-x² + 6, so let's take a look at it and see if we can do some basic analysis to figure out what properties the graph might have.
The function is a member of the family of quadratic equations since the highest degree is 2.
All quadratic equations produce a parabola as a graph.

The first term is -x². So no matter what value x has -x² will produce a negative number.
From that, we know that as x gets bigger and bigger, -x² will produce and even bigger negative number, which means the graph goes further and further down into the negative area of the graph.
From that we now know that the graph is concave down (kind of like the letter n, whereas concave up us more like the letter u). That means the graph will have a maximum point. So now our goal is to figure out what value of x produces the maximum point.

ALL values of -x² are negative. that means that we will have the case that the equation will be something like this: a negative number + 6. That means the maximum point at most can be 6 and that only happens when x=0 :: y=-x² + 6 = -0² + 6 = 0 + 6 = 6. Can you see how ANY other value of x will produce a value less than 6? (no matter what x is if it is not equal to zero it will be a negative number which means you will be taking away the negative number from 6.

With more practice, these properties will become part of what you know.
Typically, one of the first things we do is set x=0 and see what value the function produces.
Keep practicing and you will get it.
Here is a great on line graphing tool where you can experiment and get to know the properties of quadratic equations: https://www.desmos.com/calculator
• my question is that is zero a positive or a negative number? and if it is both how can we write it
(1 vote) • How would one solve a system of equations with trig functions in it? For the system: 2cos(x)+3sin(y) + (xy^3)-(3xy-1)=0 2cos(y)+9sin(x)+ 2(xy^2)-(xy-5) =0 or similar you can't use substitution. It is possible that my example has infinite solutions or no solutions , but I hope you get the idea. • Your example has infinitely many solutions.

But, in general, it is possible to find solutions to these kinds of problems, provided that you sufficiently constrain the domain.
The answers themselves will typically involve exponents with complex number, trig functions, etc. You won't typically have just a single answer.

But, this is very advanced material, far too complicated for this level of study.
• how do you notice the difference between linear and non-linear equations?
(1 vote) • It is possible to write a linear equation in the form of
`y = mx + b`
Note that this only requires that it is POSSIBLE to write a linear equation in the above fashion, not that is is currently written in that fashion. Here are some examples of other ways a linear equation might be written:
``(3y + 2)/5 = 4xy/2 + x/3 = 72y - 5x + 8 = 0y-2 = 3(x-6)y = 7(x+4)(y-5)/2 + (x-7)/3 = 0``

It is not possible to write a nonlinear equation in the above fashion. Frequently (but not always), at this level of study, a nonlinear equation is written in a form where there is a term with `x` with an exponent other than `1` OR there are factors where x is in some value of x multiplied by another value of x. Examples of nonlinear equations are
``y = 1/x + 5y = (x-7)(x+2)y = 5x² + 3y= x³ + 4x - xy= x(x+1)y = sin x``