To rewrite the system of equations as a matrix, we need to consider how the elements of the system relate to the elements of a matrix.
The equals signs in the system of equations are represented with a vertical bar in the matrix.
The coefficients of the variables form the columns on the left-hand side of the bar.
The constants form the column on the right-hand side of the bar.
Below we demonstrate this in a generalized form.
a_(11)m+ a_(12)n=c_1 a_(21)m+ a_(22)n=c_2
⇕
[
cc|c
a_(11) & a_(12) & c_1
a_(21) & a_(22) & c_2
]
When each equation in the system is written in the same order, we can consider the coefficients of the variables and the constants.
7m+ 3n=5 2m+ 1n=-1
Now that we have identified all of the variables and constants, we can place them in a matrix.
[
cc|c
7 & 3 & 5
2 & 1 & -1
]
Solving the Matrix
In order to solve the matrix, we will use row operations to obtain a matrix in the following form.
[
cc|c
1 & 0 & a
0 & 1 & b
]
This final matrix represents the solution of the system of equations, where m= a and n= b. Let's solve the matrix!
We have found that m=8 and n=-17. Now we can find x and y.
m&=1/x= 8/1 ⇒ x=1/8
n&=1/y=-17/1 ⇒ y=-1/17
The solution of the system is the point ( 18,- 117).
