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Math Review with Matlab:
Solving Algebraic
Equations
Algebraic Equations with
Multiple Variables
S. Awad, Ph.D.
M. Corless, M.S.E.E.
D. Cinpinski
E.C.E. Department
University of Michigan-Dearborn
Solving Algebraic Equations: Multiple Variables
Math Review with Matlab
U of M-Dearborn ECE Department
Solving Algebraic Equations
with Multiple Variables

Solve Command for Equations of Multiple Variables

Single Equation of Two Variables Example

Two Variable Complex Solution Example

System of Equations Example

System of Equations with Symbolic Constant Example
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Solving Algebraic Equations: Multiple Variables
Math Review with Matlab
U of M-Dearborn ECE Department
Solve Command for Equations
of Multiple Variables



The solve command can be used to solve symbolic
expressions with more than one variable:
solve(f) solves the symbolic expression f using
the symbolic variable closest to x as the
independent variable
solve(f, v) solves the symbolic expression f in
terms of the independent variable v
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Solving Algebraic Equations: Multiple Variables
Math Review with Matlab
U of M-Dearborn ECE Department
Single Equation of
Two Variables Example

Given the equation:
b  4c  2b  0
2
1) Find the solution of the equation assuming that c is
the solution variable
2) Find the solution of the equation assuming that b is
the solution variable
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Solving Algebraic Equations: Multiple Variables
Math Review with Matlab
U of M-Dearborn ECE Department
Default Solution


Solve for c as the independent variable
c is the default variable since it is closest to x
» syms b c
» f=b^2+4*c+2*b;
» sol_c=solve(f)
sol_c =
-1/4*b^2-1/2*b
b 2  4c  2b  0
2
b
b
c 
4 2
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Solving Algebraic Equations: Multiple Variables
Math Review with Matlab
U of M-Dearborn ECE Department
Solve for b


Solve for b as the independent variable
b must be explicitly specified on the command line
» sol_b=solve(f,b)
sol_b =
[ -1+(1-4*c)^(1/2)]
[ -1-(1-4*c)^(1/2)]
b  4c  2b  0
2
b1  1  1  4c
b2  1  1  4c
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Solving Algebraic Equations: Multiple Variables
Math Review with Matlab
U of M-Dearborn ECE Department
Two Variable Complex Solution
Example


Given the equation of two
variables:
x  9y  0
2
4
Solve for x in terms of y using Matlab:
» syms x y
» xs=solve(x^2+9*y^4)
xs =
[ 3*i*y^2]
[ -3*i*y^2]
Roots are
Complex Conjugates
7
Solving Algebraic Equations: Multiple Variables
Math Review with Matlab
U of M-Dearborn ECE Department
System of Equations Example

Solve the system of
consistent linear
equations:
3x1
 x2
x1
 2 x2  3x3  11
2 x1  2 x2
 2 x3 12
 x3  2
»
»
»
»
»
syms x1 x2 x3
f1='3*x1-x2+2*x3=12';
f2='x1+2*x2+3*x3=11';
f3='2*x1-2*x2+x3=2';
[x1 x2 x3]=solve(f1,f2,f3)
x1 =
7
x2 =
5
x3 =
-2
x1  7
x2  5
x3  2
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Solving Algebraic Equations: Multiple Variables
Math Review with Matlab
U of M-Dearborn ECE Department
System of Equations with
Symbolic Constant Example

Solve the two equations:


x and y are independent variables
a is a constant
x  6y  a
2x  3y  9
» syms x y a
» s=solve(x+6*y-a,2*x-3*y-9)
s =
s is a structure
x: [1x1 sym]
y: [1x1 sym]
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Solving Algebraic Equations: Multiple Variables
Math Review with Matlab
U of M-Dearborn ECE Department
Viewing Structures



The solution is a
structure with named
fields x and y
representing symbolic
expressions
View each symbolic
expression separately
Expressions are in terms
of symbolic variable a
s =
x: [1x1 sym]
y: [1x1 sym]
» xs=s.x
xs =
18/5+1/5*a
» ys=s.y
ys =
-3/5+2/15*a
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Solving Algebraic Equations: Multiple Variables
Math Review with Matlab
U of M-Dearborn ECE Department
Summary




The solve command can be used to solve:
A single equation in terms of an independent
variable
A system of multiple consistent equations
A system of equations having symbolic
variables and symbolic constants
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