Solving AlgebraicEquations

S. Awad, Ph.D.

M. Corless, M.S.E.E.

D. Cinpinski

E.C.E. Department

University of Michigan-Dearborn

Math Review with Matlab:

Algebraic Equations with Multiple Variables

U of M-Dearborn ECE DepartmentMath Review with Matlab

2

Solving Algebraic Equations: Multiple Variables

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

U of M-Dearborn ECE DepartmentMath Review with Matlab

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Solving Algebraic Equations: Multiple Variables

The solve command can be used to solve symbolic expressions with more than one variable:

Solve Command for Equations of Multiple Variables

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

U of M-Dearborn ECE DepartmentMath Review with Matlab

4

Solving Algebraic Equations: Multiple Variables

Single Equation of Two Variables Example

Given the equation: 0242 bcb

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

U of M-Dearborn ECE DepartmentMath Review with Matlab

5

Solving Algebraic Equations: Multiple Variables

Solve for c as the independent variable c is the default variable since it is closest to x

Default Solution

» syms b c» f=b^2+4*c+2*b;» sol_c=solve(f)

24

2 bbc

0242 bcb

sol_c = -1/4*b^2-1/2*b

U of M-Dearborn ECE DepartmentMath Review with Matlab

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Solving Algebraic Equations: Multiple Variables

Solve for b

» sol_b=solve(f,b)

cb

cb

411

411

2

1

Solve for b as the independent variable b must be explicitly specified on the command line

0242 bcb

sol_b =[ -1+(1-4*c)^(1/2)][ -1-(1-4*c)^(1/2)]

U of M-Dearborn ECE DepartmentMath Review with Matlab

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Solving Algebraic Equations: Multiple Variables

Given the equation of two variables:

Two Variable Complex Solution Example

09 42 yx 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

U of M-Dearborn ECE DepartmentMath Review with Matlab

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Solving Algebraic Equations: Multiple Variables

System of Equations Example

Solve the system of consistent linear equations:

2

11

12

22

32

23

321

321

321

xxx

xxx

xxx

» 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 = 7x2 = 5x3 = -2

2

5

7

3

2

1

x

x

x

U of M-Dearborn ECE DepartmentMath Review with Matlab

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Solving Algebraic Equations: Multiple Variables

System of Equations with Symbolic Constant Example

Solve the two equations:ayx 6932 yx

» syms x y a» s=solve(x+6*y-a,2*x-3*y-9)

s = x: [1x1 sym]

y: [1x1 sym]

s is a structure

x and y are independent variables a is a constant

U of M-Dearborn ECE DepartmentMath Review with Matlab

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Solving Algebraic Equations: Multiple Variables

The solution is a structure with named fields x and y representing symbolic expressions

Viewing Structures

s = x: [1x1 sym] y: [1x1 sym]

» xs=s.xxs =18/5+1/5*a

» ys=s.yys =-3/5+2/15*a

View each symbolic expression separately

Expressions are in terms of symbolic variable a

U of M-Dearborn ECE DepartmentMath Review with Matlab

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Solving Algebraic Equations: Multiple Variables

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