## 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

$\begin{array}{l}{a}_{11}{x}_{1}+{a}_{12}{x}_{2}+\dots +{a}_{1n}{x}_{n}={b}_{1}\\ {a}_{21}{x}_{1}+{a}_{22}{x}_{2}+\dots +{a}_{2n}{x}_{n}={b}_{2}\\ \cdots \\ {a}_{m1}{x}_{1}+{a}_{m2}{x}_{2}+\dots +{a}_{mn}{x}_{n}={b}_{m}\end{array}$

can be represented as the matrix equation $A\cdot \stackrel{\to }{x}=\stackrel{\to }{b}$, where A is the coefficient matrix,

$A=\left(\begin{array}{ccc}{a}_{11}& \dots & {a}_{1n}\\ ⋮& \ddots & ⋮\\ {a}_{m1}& \cdots & {a}_{mn}\end{array}\right)$

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

$\stackrel{\to }{b}=\left(\begin{array}{c}{b}_{1}\\ ⋮\\ {b}_{m}\end{array}\right)$

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.

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

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;

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 =
[  2, 1,  1]
[ -1, 1, -1]
[  1, 2,  3]

B =
2
3
-10

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

X = linsolve(A,B)
X =
3
1
-5

From X, 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.

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

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;

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]);
xSol = sol.x
ySol = sol.y
zSol = sol.z
xSol =
3
ySol =
1
zSol =
-5

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

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