if you can explain this then you pass the course prof said

[PookyBoy]

You know how you solve systems of equations like

x + 3y = 9
5x - 2y = 11

Many ways to do it, multiply the first eqn by 5 then subtract, solve for x then sub into the other eqn etc. But my prof asked us if we have ever wondered WHY we can solve systems of equations the way we do. Why can we multiply an eqn by some number or subtract one eqn from another? Does this make sense to anyone?

 

[BeefJurky]

well, you can multiply the top by 5, because you're technically multiplying both sides of an equation by 5, which doesn't change the equation.

when you subtract the one from the other, for instance if you subtracted 5x-2y=11 from the top one, you'd effectively be subtracting 11 from both sides, and not changing the equation, because 5x-2y is equal to 11.

back to my pizza. talk amongst yourselves.

 

[Rob9874]

Because the values for x and y don't change, regardless of the coefficient. Maybe I'm thinking of this too simplistically.

 

[bleeb]

You have to remember the fundamental theorm of calculus. therein lies the solution.

 

[AvesPKS]

I would say linear independence...if they weren't linearly independent, you wouldn't be able to solve this system of equations.

 

If you know that then you PASS the course. Nothing is stopping the prof from giving you a D-. Better study some too.

 

[gopunk]

just think of them as lines on a graph and it all makes sense.

 

[rgwalt]

Originally posted by: gopunk
just think of them as lines on a graph and it all makes sense.

gopunk is right. These equations are linear, and thus can be thought of a straight lines on a graph. These lines can either have no intersection, one intersection, or can lie on top of one another.

R

 

[Kyteland]

Originally posted by: PookyBoy
You know how you solve systems of equations like

x + 3y = 9
5x - 2y = 11

Many ways to do it, multiply the first eqn by 5 then subtract, solve for x then sub into the other eqn etc. But my prof asked us if we have ever wondered WHY we can solve systems of equations the way we do. Why can we multiply an eqn by some number or subtract one eqn from another? Does this make sense to anyone?

You can solve a system of linear equations like that because you are never changing the values of the variables in the equations. For instance, since you assume this equation x + 3y = 9 holsd performing the exact same mathematical operation to either side of the equation doesn't change that equality. That is why you can solve for x in that equation by subtracting 3y from both sides to get x = 9 - 3y.

Now the reason that you are able to substitute similar variables between the two equations is that you have assumed that this is a system of equations, hence the variables have the same meaning between both linear equations. You have already shown that x = 9 - 3y for the first equation so you are able to make a substitution. By making this substitution you have not changed the equality of either equation.

Now that you are limited to a single variable in a linear equation you can perform whatever mathematical maipulations are necessary to isolate the it. y=2 (someone check my math?) Again you haven't changed the equality of the equation, you have simply performed mathematical operations which preserve equality. You are now able to take the value you have computed and place it back in the original equation and get x = 3.

All other methods are a play on this same method. The key to all of them is that you assume equality in the original equation and you never perform manipulations which break that equality. Being that they are linear equations, simple mathematical operations are all that is required to isolate the variables.

 

[beer]

Linear Independence is an actual THEOREM that tells you what you are looking for.

http://www.sosmath.com/diffeq/second/linearind/linearind.html

 

[LordMorpheus]

Originally posted by: BeefJurky
well, you can multiply the top by 5, because you're technically multiplying both sides of an equation by 5, which doesn't change the equation.

when you subtract the one from the other, for instance if you subtracted 5x-2y=11 from the top one, you'd effectively be subtracting 11 from both sides, and not changing the equation, because 5x-2y is equal to 11.

back to my pizza. talk amongst yourselves.

He is right.

 

[Radiohead]

Originally posted by: BeefJurky
well, you can multiply the top by 5, because you're technically multiplying both sides of an equation by 5, which doesn't change the equation.

when you subtract the one from the other, for instance if you subtracted 5x-2y=11 from the top one, you'd effectively be subtracting 11 from both sides, and not changing the equation, because 5x-2y is equal to 11.

back to my pizza. talk amongst yourselves.

what kind of pizza you got?

 

[BamBam215]

pass a course in what? linear algebra?

 

[ed21x]

The span for the rows of a matrix isn't affected by multiplying a row by a scalar. So the range of possible answers remain constant. i hate linear algebra