How does the elimination method work for solving simultaneous equations?

[roger]

I need a math guru to explain why and how the elimination method of solving simultaneous equations works ?

why do we add or subtract the two equations ?(I undertand in order to eradicate either term) but I need to know from the basics .

For that matter, how/why does the substitution method work ?

thanks

Roger

 

[HallsofIvy]

I need a math guru to explain why and how the elimination method of solving simultaneous equations works ?

why do we add or subtract the two equations ?(I undertand in order to eradicate either term) but I need to know from the basics .

For that matter, how/why does the substitution method work ?

thanks

Roger

The answer really goes back to the nature of "=". Think of it as weighing things on balance scale. If, when you put A on one side and B on the other, if they balance, then A= B (that is, A and B weigh the same). If you add the same weight on both sides, they still balance. If you take the same weight from both sides, they still balance.

If I know that x+ y= 10 then adding x+ y to one side of any equation is the same as adding "10". In particular, if I also know that x- y= 6, then
adding x+ y to x- y gives me 2x. Adding 10 to 6 gives me 16 and that's STILL the same thing: 2x= 16 so I must have x= 8.

Long, long ago, you learned "if you do the same thing to both sides of an equation, you still have a true equation". It's still true!

 

[arildno]

roger:
Note that, just as with the scales, a balance can be sustained even if the weight on a single side has changed (as long as the weight on the other side also has changed accordingly).
It is the condition that the weights on BOTH scales are EQUAL which keeps the balance; the particular value of that shared weight has no bearing on the balancing.

 

[roger]

The answer really goes back to the nature of "=". Think of it as weighing things on balance scale. If, when you put A on one side and B on the other, if they balance, then A= B (that is, A and B weigh the same). If you add the same weight on both sides, they still balance. If you take the same weight from both sides, they still balance.

If I know that x+ y= 10 then adding x+ y to one side of any equation is the same as adding "10". In particular, if I also know that x- y= 6, then
adding x+ y to x- y gives me 2x. Adding 10 to 6 gives me 16 and that's STILL the same thing: 2x= 16 so I must have x= 8.

Long, long ago, you learned "if you do the same thing to both sides of an equation, you still have a true equation". It's still true!

What about the fact that the addition of the equation makes the y disappear, is that an optical illusion, or a false alarm ?

 

[dextercioby]

What about the fact that the addition of the equation makes the y disappear, is that an optical illusion, or a false alarm ?

Nope,it's none of them.It's something natural.Take the system of equations:
$x-y=7;y=3$.Consider the first equation:$x-y=7$.Add on both sides "y".U'll get:$x-y+y=7+y$.Reduce "y" in the LHS and you're left with:$x=7+y$.Use the second equation to get:$x=7+3=10$.
Eliminating variables/unknowns is the purpose of adding/subtracting equations.As Halls said,it is based upon the mathematical significance of the sign "=".

Daniel.

 

[fluid]

The basics:
To solve equations simultaniously you need to set them equal to each other.

It helped me to think of it graphically.
When you set two equations of 2 lines (y=mx+c) equal to each other you will find the intersecting point (the solution).

 

[roger]

why are the equations given in the form ay + bx = c ?

is the form given above strictly a function ?

 

[roger]

What I also need an explanation for is :
in dexterciobys example :

y = x-7 and y = 3

Now I set y = x-7 = 3

he said add 7 to both sides , x=10 which is the value for which f(x) is the same .
or the point where they cross,

BUT from the step, where I set the y to equal both x-7 and 3 , to the step where I find out the value of x which gives the same values for f(x) is still not intuitive in my mind.
I need someone to show me step by step ?

thanx

roger

 

[cumin]

two things that are each equal to something else are equal to each other.
If x-7=y and 3=y, then x-7=3.

 

[robert Ihnot]

Some people like to explain it this way to a class: Y=X-7 = 3
Here, we try to get under that line, and we draw a line +7=+7
Then by adding we get X=3+7 = 10.