Return to the Lessons Index | Do the Lessons in Order | Print-friendly page |
Systems of Non-Linear Equations:
Intermediate-Difficulty Systems(page 4 of 6)
Intermediate-Difficulty Systems(page 4 of 6)
Style Selections 7-3/4-in x 9-in Groutable Grecian Marble Peel and Stick Vinyl Tile. Item #1031853 Model #LSS2447HPS. Inspiring, high definition design offers contemporary style with the capability of installing with or without grout. Vinyl performance with the look of ceramic. Cheetah3D is a powerful and easy to learn 3D modeling, rendering and animation software which was developed from the ground up for Mac. So jump right into the world of computer generated imaging, create 3D artwork for your next iPhone game or make your first animated character.
The non-linear systems we've solved so far have been one quadratic equation and one linear equation, which graphed as a parabola and a straight line, respectively. Moving up in difficulty, we come to solving systems of two quadratic equations, which will graph as two parabolas; and similarly messy systems.
- Solve the following system:
y = 2x2 + 3x + 4
y = x2 + 2x + 3
y = x2 + 2x + 3
As before, I'll set these equations equal, and solve for the values of x: | 2x2 + 3x + 4 = x2 + 2x + 3 x2 + x + 1 = 0 |
Using the Quadratic Formula: | |
But I can't graph that negative inside the square root! What's going on here? Take a look at the graph: |
The lines do not intersect. Since there is no intersection, then there is no solution. That is, this is an inconsistent system. My final answer is: no solution: inconsistent system.
In general, the method of solution for general systems of equations is to solve one of the equations (you choose which) for one of the variables (again, you choose which). Then you plug the resulting expression into the other equation for the chosen variable, and solve for the values of the other variable. Then you plug those solutions back into the first equation, and solve for the values of the first variable. Here are some additional examples: Copyright © 2002-2011 Elizabeth Stapel All Rights Reserved
- Solve the following system:
y = –x – 3
x2 + y2 = 17
x2 + y2 = 17
Graphically, this system is a straight line (from the first equation) crossing a circle centered at the origin (from the second equation):
There appear to be two solutions. I'll proceed algebraically to confirm this impression, and to get the exact values.
Since the first equation is already solved for y, I will plug '–x – 3' in for 'y' in the second equation, and solve for the values of x:
x2 + y2 = 17
x2 + (–x – 3)2 = 17
x2 + (–x – 3)(–x – 3) = 17
x2 + (x2 + 6x + 9) = 17
2x2 + 6x + 9 = 17
2x2 + 6x – 8 = 0
x2 + 3x – 4 = 0
(x + 4)(x – 1) = 0
x = –4, x = 1
x2 + (–x – 3)2 = 17
x2 + (–x – 3)(–x – 3) = 17
x2 + (x2 + 6x + 9) = 17
2x2 + 6x + 9 = 17
2x2 + 6x – 8 = 0
x2 + 3x – 4 = 0
(x + 4)(x – 1) = 0
x = –4, x = 1
When x = –4, y = –x – 3 = –(–4) – 3 = 4 – 3 = 1
When x = 1, y = –x – 3 = –(1) – 3 = –4
Then the solution consists of the points (–4, 1) and (1, –4).
Note the procedure: I solved one of the equations (the first equation looked easier) for one of the variables (solving for 'y=' looked easier), and then plugged the resulting expression back into the other equation. This gave me one equation in one variable (the variable happened to be x), and a one-variable equation is something I know how to solve. Once I had the solution values for x, I back-solved for the correspondingy-values. I emphasize 'corresponding' because you have to keep track of which y-value goes with which x-value. In the example above, the points (–4, –4) and (1, 1) are not solutions. Even though I came up with x = –4 and 1 and y = –4 and 1, the x = –4 did not go with the y = –4, and the x = 1 did not go with the y = 1.
Warning: You must match the x-values and y-values correctly. Gone home 1 1 download free. Be careful!
Content Continues Below
- Solve the following system of equations:
y = (1/2)x – 5
y = x2 + 2x – 15
y = x2 + 2x – 15
Since both equations are already solved for y, I'll set them equal and solve for the values of x:
(1/2)x – 5 = x2 + 2x – 15
x – 10 = 2x2 + 4x – 30
0 = 2x2 + 3x – 20
0 = (2x – 5)(x + 4)
x – 10 = 2x2 + 4x – 30
0 = 2x2 + 3x – 20
0 = (2x – 5)(x + 4)
x = 5/2, x = –4
When x = 5/2:
y = (1/2)x – 5 = (1/2)(5/2) – 5 = 5/4 – 20/4 = – 15/4 = –3.75
When x = –4:
y = (1/2)x – 5 = (1/2)(–4) – 5 = –2 – 5 = –7
Then the solutions are the points ( 5/2, –15/4 ) and (–4, –7).
Graphically, the above system looks like this: The intersection points on the graph appear to be good matches for the numerical solutions I got via algebra, confirming that I've done the exercise correctly. |
- Solve the following system of equations:
xy = 1
x + y = 2
x + y = 2
Taking a quick look at the graph, I see that there appears to be only one solution:
I guess I'll solve the second equation for y, and plug the result into the first equation:
x + y = 2
y = –x + 2 Netnewswire 4 1 0 – rss and atom news reader.
y = –x + 2 Netnewswire 4 1 0 – rss and atom news reader.
Then:
xy = 1
x(–x + 2) = 1
–x2 + 2x = 1
–x2 + 2x – 1 = 0
x2 – 2x + 1 = 0
(x – 1)(x – 1) = 0
x = 1
x(–x + 2) = 1
–x2 + 2x = 1
–x2 + 2x – 1 = 0
x2 – 2x + 1 = 0
(x – 1)(x – 1) = 0
x = 1
Then:
x + y = 2
(1) + y = 2
y = 1
(1) + y = 2
y = 1
Then the solution is the point(1, 1).
<< PreviousTop |1 | 2 | 3 | 4 | 5 | 6| Return to IndexNext >>
Cite this article as: | Stapel, Elizabeth. 'Systems of Non-Linear Equations: Intermediate Systems.' Purplemath. Available from https://www.purplemath.com/modules/syseqgen4.htm. Accessed [Date] [Month] 2016 |
Cheetah3D on BBC iPlayer →
Cheetah3D creator Martin Wengenmayer appeared on the Cheetah3D forum recently and listed some of the features that we can expect to see in the upcoming Cheetah3D Version 7. Whilst there’s no release date yet, the new feature list does sound impressive…
- New renderer
- NGon booleans
- Movie textures
- Soft selections
- Collada import
- Layers
- Big UI update
- and much much more
There’s a few in there that I’m pretty excited about, and I must confess I do know of some other features that are planned too but I’m sworn to secrecy on them!
New Renderer
My knowledge of rendering isn’t the best and I’m not sure what to expect here. Could we finally see the much requested render preview?
NGon Booleans
This is a big one. Booleans have never been particularly useful in Cheetah3D as the meshes that you end up with are quite messy. It’s very hard to move on and continue modelling after you’ve used a boolean at present. All this is set to change though. I can actually see this changing the way I model completely.
Movie Textures
Personally I’ve never really needed movie textures. Tangledeep 1 23e (27495) download free. I think this is because I largely produce still images with Cheetah3D. I can see a huge benefit here though and it’s something I’ve seen requested in the forum a good few times over the years. The possibilities here seem pretty endless. Initially I was just thinking about things like TV screens, but actually it could go way further than this.
Soft Selections
Soft selections is something I’ve personally requested. I first encountered them in Silo and what a great feature it is. If you’ve never used it, it’s quite hard to explain but I’ll try anyway:- Soft selection allows you to select part of a model and then take an additional softer selection of it’s surrounding points. The softer selection has less influence but whatever you do (move, scale) still has an impact. It’s an absolute must if you’re doing any organic modelling.
Layers
Probably the most intriguing feature on the list. I’ve not seen layers in any 3D application before. I’m not saying they don’t exist but it’s not something that I’ve used and I struggle to even understand how it would work. I’ve seen rendering to layers before in Strata but I don’t think this is what Martin is talking about. If anyone can enlighten me, I’d really appreciate it!
UI Update
Cheetah3d 7 3 3 X 4 X 6 4
A UI update is a scary thing for me! Does it mean that my tutorial that are all done in v5 and 6 will need updating? I have no idea at this stage.
Cheetah3d 7 3 3 X 4 Area Rugs
![Cheetah3d 7 3 3 X 4 Cheetah3d 7 3 3 X 4](https://media.cgtrader.com/variants/BEZSBnPt4UrihSjUoS5x6UR3/c9bd63b7c369a06d872dbd8b861d5c546820c3b0b13ae1df404f621385bf58ca/Jesus in cross-J45.jpg)
That’s my thoughts. I’ve not mentioned the Collada import as I have no idea what Collada is really used for. Again I’m all ears if anyone wants to tell me!