Simultaneous Linear Equations

This lesson explains how to use matrix methods to represent simultaneous linear equations compactly and to solve equations efficiently.

How to Represent Simultaneous Linear Equations In Matrix Form

Suppose you have n linear equations with n unknowns. Using ordinary algebra, those equations might be expressed as:

A₁₁x₁ + A₁₂x₂ + A₁₃x₃ + . . . + A_1nx_n = y₁
A₂₁x₁ + A₂₂x₂ + A₂₃x₃ + . . . + A_2nx_n = y₂
A₃₁x₁ + A₃₂x₂ + A₃₃x₃ + . . . + A_3nx_n = y₃
. . .
A_n1x₁ + A_n2x₂ + A_n3x₃ + . . . + A_nnx_n = y_n

where

x_j is an unknown value
A_ij is the known coefficient of x_j in equation i
y_j is a known quantity in equation j

This set of equations can be expressed compactly in matrix form as follows:

Ax = y

where

x is an n x 1 column vector of unknown values x₁, x₂, . . . , x_n
A is an n x n matrix of the known coefficients A_ij
y is an n x 1 column vector of known values y₁, y₂, . . . , y_n

How to Solve Simultaneous Linear Equations Using Matrix Methods

Here is how to solve a system of n linear equations in n unknowns, using matrix methods.

• Express the set of n linear equations compactly in matrix form.

  Ax = y

• Premultiply both sides of the equation by A^-1, the inverse of A.

  A^-1Ax = A^-1y

• Since A^-1Ax = Ix = x, we know the following.

  x = A^-1y

Thus, as long as the inverse A^-1 exists, we can solve for x, the vector of unknown values. If the inverse does not exist, the set of equations does not have a unique solution.

Solving Simultaneous Linear Equations: An Example

To illustrate how to solve simultaneous linear equations using matrix methods, consider the following set of equations.

  x₁ + 2x₂ + 2x₃ = 1
2x₁ + 2x₂ + 2x₃ = 2
  2x₁ + 2x₂ + x₃ = 3

We want to solve for the unknown quantities: x₁, x₂, and x₃.

• Our first step is to express these equations in matrix form as Ax = y.

• Next, we premultiply both sides of the equation by A^-1, the inverse of matrix A. Previously, we showed how to find the inverse of matrix A.

• And finally, since A^-1Ax = Ix = x, we know the following.

Thus, we have solved for the unknown quantities: x₁ = 1, x₂ = 1, and x₃ = -1.

Test Your Understanding of This Lesson

Problem 1

Consider the following set of simultaneous linear equations.

  3x₁ + x₂ = 3
9x₁ + 4x₂ = 6

Using matrix methods, solve for the unknown quantities: x₁and x₂.

Solution

Our solution involves a three-step process.

• The first step is to express these equations in matrix form as Ax = y.

• Next, we premultiply both sides of the equation by A^-1, the inverse of matrix A. We showed how to find the inverse of matrix A in a previous lesson.

• And finally, since A^-1Ax = Ix = x, we know the following.

Thus, we have solved for the unknown quantities: x₁ = 2 and x₂ = -3.