How come I can add two equations?
Say we've got
$\displaystyle 2x+y=3\\-2x+y=4$
I can add the functions to get:
$\displaystyle 2y=7$
What is the reasoning behind this? Thanks!
Can I presume that you know that, for any a, b, and c, "if a= b then a+ c= b+ c"?
Okay, if a= b and c= d then, first, adding c to both sides of the first equality, a+ c= b+ c. And, second, adding b to both sides of the second equality, b+ c= b+ d.
Of course, if X= Y and Y= Z then X= Z. Since a+ c= b+ c and b+ c= b+ d, then a+ c= b+ d.
Your question is about the reason for adding the two equations, indeed it could be a subtraction as well. For that you need to understand this:
If you have two equation with two unknowns x and y and you need to find the values of the unknowns, one of the method is the elimination. It works like this:
(1) ax + by = c
(2) -ax + dy = e
You will need to reduce the two equations into one equation. For that you will have to eliminate one of the two equations. Notice that if you add (1) to (2) term by term, the coefficents of x are the same with opposite signs. The addition results in this
ax + (-a)x + by + dy = c + e
You see here the x terms cancel out and you will be left with (b + d)y = (c + e) ==> y = (c + e)/(b + d)
Now you know the value of y, replace it in either (1) or (2), you will end up with one equation with x, which I understand you could solve for x without difficulty.
You should notice that it is not alway addition of the equations that leads to the solution. Sometimes it is a subtraction. here is an example:
(1) ax + by = c
(2) ax + dy = e
Here the coefficients of x are the same and both are positive (they could be both neagative). If you add the two equations you will and up with one equation with two unknowns. Do it and see what you get.
You cannot solve for two unknowns with one equation. But if you subtract, say (2) from (1) you will have:
ax - ax + by - dy = c - d <---- compare the negative sign with the + sign in the previous example.
Now you can solve for y then use it in either (1) or (2) to solve for x.
The bottom line is: the reason for addition, or subtraction of two equations with two unknown is to reduce the two equations into one equation with one unknown. This is one of the methods for solving two linear equations.
We can only add in this case because we have the same Leading Coefficient in both equations. Note how the L.C is negative in the second equation, so when we add both equations , the x terms will cancel each other out, leaving behind 2y = 7.
For example if we have: 5x + 2y = 10 (1)
3x + 2y = 4 (2)
In this case both y terms are the same, so how can we get rid of them? We simply subtract in this case (1) - (2) or (2)-(1). So we end up with 2x = 6 , x= 3 or -2x = -6 , x=3 (By the way , you can only subtract like terms )