Question 1205195: what is the answer to this elimination equation
-4x+2y=8
8x-4y=-16
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Given system
-4x+2y=8
8x-4y=-16
Double both sides of the 1st equation to end up with -8x+4y=16
So the given system is equivalent to this system
-8x+4y=16
8x-4y=-16
Add straight down.
The x's cancel.
The y's cancel.
The right hand sides add to 0.
We end up with 0x+0y = 0 or simply 0 = 0.
The equation 0 = 0 is always true.
There are infinitely solutions.
Each solution is of the form (x,y) where y = 2x+4
y = 2x+4 is what happens when you solve for y in any of the equations mentioned.
We could say that each solution is of the form (x, 2x+4) to avoid involving another variable.
If x = 0, then we get (x, 2x+4) = (0, 2*0+4) = (0, 4)
If x = 1, then we get (x, 2x+4) = (1, 2*1+4) = (1, 6)
If x = 2, then we get (x, 2x+4) = (2, 2*2+4) = (2, 8)
If x = 3, then we get (x, 2x+4) = (3, 2*3+4) = (3, 10)
and so on.
If you used a tool like Desmos or GeoGebra to graph, then you should notice that the two original equations produce the exact same line. Click one equation to turn it off, then click it back on, repeating this process to see the line blink. It should help show the overlap.
If you prefer something like a TI83, then stick to that.
Answer by ikleyn(52890)  (Show Source):
You can put this solution on YOUR website! .
Multiply first equation by 2 and add to the second equation.
After canceling like terms, you will get then an IDENTITY
0 = 0,
which is valid ALWAYS at any values of x and y.
It means that two lines, defined by the two equations, coincide,
which means that the two equations are equivalent and the system has infinitely many solutions.
You can take any value of x and from equations find corresponding value of y.
The points (x,y) obtained this way, are the solutions to the system.
These points belong to these two coinciding straight lines.
Solved, with explanations.
See the lesson
- Geometric interpretation of the linear system of two equations in two unknowns
in this site.
//////////////////
Wording in this post and in other your two posts is TERRIBLE.
It is only good to scare people around.
A correct wording is THIS:
Solve the system of two equations in two unknowns by the elimination method.
On the elimination method, see the lesson
- Solution of the linear system of two equations in two unknowns by the Elimination method
in this site.
Learn the method and the relevant terminology from my lessons in order to
communicate with people around in a right language.
|
|
|
