How to write a system of equations as an augmented matrix
An augmented matrix keeps the coefficients and constants from a linear system in a compact array. Each row represents an equation, and each variable gets its own column. We’ll line up the equations first so every entry has a clear place.
Use a column for each variable and one for constants
The first column contains the coefficients of x, the second contains the coefficients of y, and the final column contains the constants. The vertical divider separates the coefficient part from the right side.
Compare coefficient and augmented matrices
For the system and , use the coefficients for the first column and the coefficients for the second. That gives the coefficient matrix.
Use zero for a missing variable
Choose the order . The first equation has no term, so its coefficient in the third column is zero.
Two equations give two rows. Four variables plus a constant column give five columns, so this augmented matrix is .
Rearrange equations before taking their coefficients
Put the variables in the order on the left and the constants on the right.
Read the matrix back as equations
The third row represents . Reading each row back checks that the signs, zero coefficients, and right-side constants match the system. The matrix records the system; row operations are the next step if we want to solve it.