HP 15c User Manual

Page 17

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Section 1: Using

_ Effectively

17

For some systems of equations, expressed as

f

1

(x

1

, …, x

n

) = 0

f

n

(x

1

, …, x

n

) = 0

it is possible through algebraic manipulation to eliminate all but one variable. That is, you
can use the equations to derive expressions for all but one variable in terms of the remaining
variable. By using these expressions, you can reduce the problem to using _ to find
the root of a single equation. The values of the other variables at the solution can then be
calculated using the derived expressions.

This is often useful for solving a complex equation for a complex root. For such a problem,
the complex equation can be expressed as two real-valued equations—one for the real
component and one for the imaginary component—with two real variables—representing the
real and imaginary parts of the complex root.

For example, the complex equation z + 9 + 8e

−z

= 0 has no real roots z, but it has infinitely

many complex roots z = x + iy. This equation can be expressed as two real equations

x + 9 + 8e

−x

cos y = 0

y − 8e

x

sin y = 0.

The following manipulations can be used to eliminate y from the equations. Because the sign
of y doesn't matter in the equations, assume y > 0, so that any solution (x,y) gives another
solution (x,−y). Rewrite the second equation as

x = ln(8(sin y)/y),

which requires that sin y > 0, so that 2< y < (2n + 1)π for integer n = 0, 1, ....

From the first equation

y = cos

−1

(−e

x

(x + 9)/8) + 2

= (2n + 1)π − cos

−1

(e

x

(x + 9)/8)

for n = 0, 1, … substitute this expression into the second equation,

0

))

9

(

(

64

)

8

/

)

9

(

(

cos

)

1

2

(

ln

2

1



x

e

x

e

n

x

x

x

.

You can then use _ to find the root x of this equation (for any given value of n, the
number of the root). Knowing x, you can calculate the corresponding value of y.

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