How do matrices represent systems of equations?

If we have a system of two equations:

$$\begin{array}{r}{a}_1{x}_1+a_2{x}_2+...a_n{x}_n=c_1\\ b_1{x}_1+b_2{x}_2+...b_n{x}_n=c_2\end{array}$$

• We use elimination method to solve the equation.
• Elimination method is basically just used to cancel out the coefficients, so that we can see what the variables equate to.
• Since we are primarily operating on the coefficients, it would be convenient to not have to deal with writing down the variables.

$$\left[\begin{array}{cccc}{a}_1& a_2& ...& a_n\\ b_1& b_2& ...& b_n\end{array}\right]$$

• As long as we maintain the placement of each of the coefficients in the matrix, operating on this matrix is same as operating on the equations.
• Since we also have an RHS with the constant term, we use something called an augmented matrix.

$$\left[\begin{array}{ccccc}{a}_1& a_2& ...& a_n& c_1\\ b_1& b_2& ...& b_n& c_2\end{array}\right]$$

This is just an alternate (more convenient) representation of the equations.