Simultaneous Linear Equations
This lesson explains how to use matrix methods to (1) represent a system of linear equations compactly and (2) solve simulataneous linear equations efficiently.
How to Represent a System of Linear Equations In Matrix Form
Suppose you have n linear equations with n unknowns. Using ordinary algebra, those equations might be expressed as:
A_{1}_{1}x_{1} +
A_{1}_{2}x_{2} +
A_{1}_{3}x_{3} + . . . +
A_{1}_{n}x_{n} = y_{1}
A_{2}_{1}x_{1} +
A_{2}_{2}x_{2} +
A_{2}_{3}x_{3} + . . . +
A_{2}_{n}x_{n} = y_{2}
A_{3}_{1}x_{1} +
A_{3}_{2}x_{2} +
A_{3}_{3}x_{3} + . . . +
A_{3}_{n}x_{n} = y_{3}
. . .
A_{n}_{1}x_{1} +
A_{n}_{2}x_{2} +
A_{n}_{3}x_{3} + . . . +
A_{n}_{n}x_{n} = y_{n}
where
x_{j} is an unknown value
A_{i}_{j} 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 x_{1}, x_{2}, . . . ,
x_{n}
A is an n x n matrix of the known
coefficients A_{i}_{j}
y is an n x 1 column vector
of known values y_{1}, y_{2}, . . . ,
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^{1}Ax = A^{1}y

Since A^{1}Ax = Ix = x, we know the following.
x = A^{1}y
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 system of equations.
x_{1} + 2x_{2} + 2x_{3} = 1
2x_{1} + 2x_{2} + 2x_{3} = 2
2x_{1} + 2x_{2} + x_{3} = 3
We want to solve for the unknown quantities: x_{1}, x_{2}, and x_{3}.
 Our first step is to express these equations in matrix form as Ax = y.


= 


A  x  y 
 Next, we premultiply both sides of the equation by
A^{1}, the inverse of matrix
A. This results in the following relationship.
A^{1}Ax = A^{1}y
Recall that we showed how to find the inverse of matrix A in a previous lesson.
 And finally, since A^{1}Ax = Ix = x, we know that x = A^{1}y. Thus,

= 


= 


x  A^{1}  y 
Thus, we have solved for the unknown quantities: x_{1} = 1, x_{2} = 1, and x_{3} = 1.
Test Your Understanding
Problem 1
Consider the following system of linear equations.
3x_{1} + x_{2} = 3
9x_{1} + 4x_{2} = 6
Using matrix methods, solve for the unknown quantities: x_{1}and x_{2}.
Solution
Our solution involves a threestep process.
 The first step is to express these equations in matrix form as Ax = y.


= 


A  x  y 
 Next, we premultiply both sides of the equation by
A^{1}, the inverse of matrix
A. This results in the following relationship.
A^{1}Ax = A^{1}y
Recall that we showed how to find the inverse of matrix A in a previous lesson.
 And finally, since A^{1}Ax = Ix = x, we know the following.

= 


= 


x  A^{1}  y 
Thus, we have solved for the unknown quantities: x_{1} = 2 and x_{2} = 3.