Systems of Three Equations: Solving using Matrices and Row Reduction

Systems of Three Equations Solving using Matrices and Row Reduction

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Solving using Matrices and Row Reduction

Systems with three equations and three variables can also be solved using matrices and row reduction. First, arrange the system in the following form:

a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃

where a_{1, 2, 3}, b_{1, 2, 3}, and c_{1, 2, 3} are the x, y, and z coefficients, respectively, and d_{1, 2, 3} are constants.

Next, create a 3×4 matrix, placing the x coefficients in the 1st column, the y coefficients in the 2nd column, the z coefficients in the 3rd column, and the constants in the 4th column, with a line separating the 3rd column and the 4th column:

This is equivalent to writing

which is equivalent to the original three equations (check the multiplication yourself).

Finally, row reduce the 3×4 matrix using the elementary row operations. The result should be the identity matrix on the left side of the line and a column of constants on the right side of the line. These constants are the solution to the system of equations: