What is the Proof of the addition or elimination method for solving systems of equations in two variables like x and y. No Z variable included.

Question

What is the proof for the addition or elimination method

Answer 1

It actually has to do with linear algebra and how we consider systems of equations with matrices. To start, I'll go through a basic system with substitution, elimination, and finally, matrices.

1. X+3y=7

2. -X+4y=7

Note: I labeled the equations as equation 1 and equation 2.

Substitution

We rearrange eq 1 to solve for either variable. In this case, I'll solve for X

X=7-3y

I will now substitute 7-3y for X in eq 2.

-(7-3y)+4y=7

-7+3y+4y=7

7y=14

y=2

Using this value of y in either equation allows us to find X.

X=7-3(2)

X=1

Elimination

Now, we will add the equations together in order to eliminate a variable from the system. We can multiply the equations by a constant but in this case, we don't need to. Adding eq1 and eq2 together gives us

7y=14

y=2

Same as before, X=1

Row Operations

This is the linear algebra I was talking about that uses matrices. Please note that there are rules for how matrices can multiply, row by column. We can use this property to represent a system under a single equation. Also note that a comma in a matrix is a new row under what came before. Knowing this, we can represent our system as:

[1 3 , -1 4][X , y] = [7 , 7]

The first matrix is the coefficient matrix and you will notice that the first line contains the coefficients of eq1 and the second of eq 2. Row operations are a useful tool we can use to manipulate one row at a time in order to get the coefficient matrix to be [1 0 , 0 1]. For ease, we can represent the matrix as [1 3 | 7, -1 4 | 7]. This is to make it easier to manipulate the rows. If we add Row 1 to Row 2 and store it in Row 2, we will get [1 3 | 7, 0 7 | 14]. Then we can divide Row 2 by 7 to get [1 3 | 7, 0 1 | 2]. Finally, we can subtract 3 times Row 2 from Row 1 and store it in Row 1. [1 0 | 1, 0 1 | 2]. This is starting to look familiar but I will rewrite it in standard form.

[1 0, 0 1][X, y] = [1, 2]

In the language of matrices, this represents x=1,y=2.

I don't know if this helps explain why it works but I hope I've shown you how to do similar operations in a different way.