How to write a system of linear equations as an augmented matrix

Key takeaway: A system in augmented-matrix form is just a compact way to line up coefficients and constants. The key is to rewrite every equation so the variables appear in the same order before you place the numbers into rows.

Let the variables be ordered as $x, y, z$.

1) Rewrite each equation in standard form

Given

$x - 2z + 4y = 6$

write it as

$x + 4y - 2z = 6.$

Next,

$x = 2y + 7$

becomes

$x - 2y + 0z = 7.$

And

$z = -7x + 2y$

becomes

$7x - 2y + z = 0.$

2) Read off the coefficients

Now take the coefficients of $x, y, z$ and the constants:

1 & 4 & -2 & 6 \\ 1 & -2 & 0 & 7 \\ 7 & -2 & 1 & 0 \end{bmatrix}$$ That is the augmented matrix. ### 3) Quick check Each row matches one equation, and the last column holds the constants on the right-hand side. --- **Pitfalls the pros know** 👇 A common mistake is to copy the equations in the order they are written instead of first putting them in the same variable order. For example, $x - 2z + 4y = 6$ must be rewritten as $x + 4y - 2z = 6$ before you build the matrix. Also watch the signs when moving terms across the equals sign. **What if the problem changes?** If the variables were ordered as $z, y, x$ instead, the matrix would look different but still represent the same system. The important thing is consistency: choose one variable order and use it for every row. If a variable is missing from an equation, its coefficient is $0$. `Tags`: coefficient matrix, augmented matrix, row form