Rearrange each equation, if necessary, so all the variables are on one side of the equation and the constant is on the other side. If one of the equations is 2x + y - 3z = 15 - 3y, for example, add 3y to both sides of the equation to get 2x + 4y - 3z = 15.
Add zero times the variable to an equation if the equation does not have one of the variables used in the system. If a system has the variables x, y and z but one of the equations only has y - 3z = 15 and no x-variable, for example, write the equation as 0x + y - 3z = 15. Place the variables in the same order in each equation -- for example, the x-variable, then the y-variable, then the z-variable.
Draw a left-side square bracket -- the symbol "[" -- long enough to contain as many horizontal rows as there are equations.
Write the coefficients for each variable in the first equation in the top row of the matrix. Make sure to include any zeros. If the first equation is 0x + y - 3z = 15, for example, the first row will contain 0, 1 and -3, in that order.
Write the coefficients of the variables in the second equation in the second row of the matrix. If the second equation is 2x +4y - 3z = 15, for example, the second row will contain 2, 4 and -3, in that order. Continue writing the coefficients of each equation in a new row until you have added all the equations to the matrix.
Draw a vertical line after the last column of coefficients.
Write the constant from the right-hand side of the first equation in the top row of the matrix after the vertical line. Write the constant from the second equation in the second row. If the first equation is 0x + y - 3z = 5 and the second equation is 2x + 4y - 3z = 15, for example, write 5 in the first row after the line and 15 in the second row. Continue in this manner until you have added the constants from each equation to the matrix.
Close the augmented matrix with a right-hand square bracket: "]".