Solve System of Linear Equations

This example shows 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

$$\begin{array}{l}{\mathit{a}}_{11}{\mathit{x}}_1+{\mathit{a}}_{12}{\mathit{x}}_2+\dots +{\mathit{a}}_{1\mathit{n}}{\mathit{x}}_{\mathit{n}}={\mathit{b}}_1\\ {\mathit{a}}_{21}{\mathit{x}}_1+{\mathit{a}}_{22}{\mathit{x}}_2+\dots +{\mathit{a}}_{2\mathit{n}}{\mathit{x}}_{\mathit{n}}={\mathit{b}}_2\\ \dots \\ {\mathit{a}}_{\mathit{m}1}{\mathit{x}}_1+{\mathit{a}}_{\mathit{m}2}{\mathit{x}}_2+\dots +{\mathit{a}}_{\mathit{mn}}{\mathit{x}}_{\mathit{n}}={\mathit{b}}_{\mathit{m}}\end{array}$$

can be represented as the matrix equation $\mathit{A}\overrightarrow{\mathit{x}}=\overrightarrow{\mathit{b}}$, where $$A$$ is the coefficient matrix

$$\mathit{A}=\left[\begin{array}{ccc}{\mathit{a}}_{11}& \cdots & {\mathit{a}}_{1\mathit{n}}\\ ⋮& \ddots & ⋮\\ {\mathit{a}}_{\mathit{m1}}& \cdots & {\mathit{a}}_{\mathit{mn}}\end{array}\right]$$

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

$$\overrightarrow{\mathit{b}}=\left[\begin{array}{c}{\mathit{b}}_1\\ ⋮\\ {\mathit{b}}_{\mathit{m}}\end{array}\right]$$

If you do not have the system of linear equations in the form AX = B, use equationsToMatrix to convert the equations into this form. Consider the following system.

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

Define 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;

Use equationsToMatrix to convert the equations into the form AX = B. The second input to equationsToMatrix specifies the independent variables in the equations.

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

A = 
$$\left(\begin{array}{ccc}2& 1& 1\\ -1& 1& -1\\ 1& 2& 3\end{array}\right)$$

B = 
$$\left(\begin{array}{c}2\\ 3\\ -10\end{array}\right)$$

Use linsolve to solve AX = B for the vector of unknowns X.

X = linsolve(A,B)

X = 
$$\left(\begin{array}{c}3\\ 1\\ -5\end{array}\right)$$

From the result in X, the solutions of the system are $$x=3$$, $$y=1$$, and $$z=-5$$.

Solve System of Linear Equations Using solve

Use solve instead of linsolve if you have the equations in the form of expressions and not a matrix of coefficients. Consider the same system of linear equations.

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

Define 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;

Solve the system of equations using solve. The inputs to solve are a vector of equations, and a vector of variables to solve the equations for.

sol = solve([eqn1,eqn2,eqn3],[x,y,z]);

solve returns the solutions in a structure array. To access the solutions, index into the array.

xSol = sol.x

xSol = $$3$$

ySol = sol.y

ySol = $$1$$

zSol = sol.z

zSol = $$-5$$

Related Topics

• Solve Algebraic Equations
• Solve System of Algebraic Equations