Solve System of Linear Equations

This section shows you how to solve a system of linear equations using the Symbolic Math Toolbox™.

Solve System of Linear Equations Using linsolve

A system of linear equations

$\begin{array}{l} a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 n} x_{n} = b_{1} \\ a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 n} x_{n} = b_{2} \\ \dots \\ a_{m 1} x_{1} + a_{m 2} x_{2} + \dots + a_{m n} x_{n} = b_{m} \end{array}$

can be represented as the matrix equation $A \cdot \vec{x} = \vec{b}$ , whereAis the coefficient matrix,

$A = (\begin{matrix} a_{11} & \dots & a_{1 n} \\ ⋮ & ⋱ & ⋮ \\ a_{m 1} & \dots & a_{m n} \end{matrix})$

and $\vec{b}$ is the vector containing the right sides of equations,

$\vec{b} = (\begin{matrix} b_{1} \\ ⋮ \\ b_{m} \end{matrix})$

If you do not have the system of linear equations in the formAX = B, useequationsToMatrixto convert the equations into this form. Consider the following system.

$\begin{array}{l} 2 x + y + z = 2 \\ - x + y - z = 3 \\ x + 2 y + 3 z = - 10 \end{array}$

Declare the system of equations.

syms x y z eqn1 = 2*x + y + z == 2; eqn2 = -x + y - z == 3; eqn3 = x + 2*y + 3*z == -10;

UseequationsToMatrixto convert the equations into the formAX = B. The second input toequationsToMatrixspecifies the independent variables in the equations.

[A,B] = equationsToMatrix([eqn1, eqn2, eqn3], [x, y, z])

A = [ 2, 1, 1] [ -1, 1, -1] [ 1, 2, 3] B = 2 3 -10

Uselinsolveto solveAX = Bfor the vector of unknownsX.

X = linsolve(A,B)

X = 3 1 -5

FromX,x= 3,y= 1andz= -5.