Question about basic concept of solving system of equations

[lustrog]

why does adding equations solve the system of equations? I've taken matrix algebra but, to be honest, was too bashful to ask such a simple question. :P

 

[Mark44]

You can always add the same number to both sides of an equation, to get a new equation with the same solution set (an equivalent equation).

For example, if your equation is x - 2 = 5, you can add 2 to both sides, getting x = 7.

For the same reason you can add one side of an equation to one side of a 2nd equation, and add the other side of the first equation to the 2nd equation. Since both sides of an equation represent the same number, you are adding the same quantity (in different forms) to another equation.

Suppose you have this system:
x + y = 3 (eqn 1)
x - y = 1 (eqn 2)

If I add the left side of equation 1 to the left side of equation 2, and the right side of equation 1 to the right side of equation 2, I get this equation:
2x = 4, from which I get x = 2. I can then use either of my original equations to solve for y, which of course is 1.

You can extend this idea to adding a multiple of one equation to another. The equation being added states that two quantities are equal. I can multiply both sides of that equation by whatever I want (other than 0) to get an equivalent equation. The revised equation can be added to another equation, just as in the previous example.

 

[lustrog]

much appreciated, mark44. one of the things that bothered me is why you can treat the x's and y's of each equation as the same value. but your response put me on the right track to clearing that up too, because if you think about it in two steps it makes sense.

(if anyone else has been bothered by this also, here is the way I'm thinking about it now):
if you take y's equal at any given point, but let the x's from each equation, x_1, and x_2 be anything, then you can add the two equations together to find that when y's are equal, you have x_1 + x_2 = 4. but we're interested when it is also the case that x_1 = x_2, so we have that 2*x = 4 and x = 2.

i don't know why i sometimes get hung up on such things when others don't but this was very helpful. thanks again.

 

[HallsofIvy]

much appreciated, mark44. one of the things that bothered me is why you can treat the x's and y's of each equation as the same value. but your response put me on the right track to clearing that up too, because if you think about it in two steps it makes sense.

Because they are the same value! That's exactly what the "simultaneous" in "simultaneous equations" means- find specific values of x, y, etc. that satisfy all of the equations.

(if anyone else has been bothered by this also, here is the way I'm thinking about it now):
if you take y's equal at any given point, but let the x's from each equation, x_1, and x_2 be anything, then you can add the two equations together to find that when y's are equal, you have x_1 + x_2 = 4. but we're interested when it is also the case that x_1 = x_2, so we have that 2*x = 4 and x = 2.

i don't know why i sometimes get hung up on such things when others don't but this was very helpful. thanks again.

 

[lustrog]

lol. yes i guess it really is that simple. thanks for helping me along though.

 

[paulfr]

why does adding equations solve the system of equations? I've taken matrix algebra but, to be honest, was too bashful to ask such a simple question. :P

To directly answer your question you do it [aadding or subtracting] because it eliminates one of the two variables and leads to the value for the other variable. Then substitution back to either original equation yields the other value.