Example 01 - Simultaneous Non-Linear Equations of Three Unknowns

Problem
Solve for x, y, and z from the following simultaneous equations.

$z^x \, z^y = 100\,000$ ← equation (1)

$(z^x)^y = 100\,000$ ← equation (2)

$\dfrac{z^x}{z^y} = 10$ ← equation (3)

Solution

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Multiply equation (1) and equation (3)
$(z^x \, z^y)\left( \dfrac{z^x}{z^y} \right) = 100\,000(10)$

$z^{2x} = 1\,000\,000$

$z^x = 1000$

Substitute z^x = 1000 to equation (2)
$1000^y = 100\,000$

$10^{3y} = 10^5$

Thus,
$3y = 5$

$y = \frac{5}{3}$

Another way to solve for y †

Solving for y using logarithm

Substitute z^x = 1000 to equation (2)
$1000^y = 100\,000$

$\log 1000^y = \log 100\,000$

$y \, \log 1000 = \log 100\,000$

$y = \dfrac{\log 100\,000}{\log 1000}$

$y = \frac{5}{3}$

Divide equation (1) by equation (3)
$\dfrac{z^x \, z^y}{\dfrac{z^x}{z^y}} = \dfrac{100\,000}{10}$

$z^{2y} = 10\,000$

$z^y = 100$

Substitute y = 5/3 to z^y = 100 from above
$z^{5/3} = 100$

$z = 100^{3/5}$

Substitute y = 5/3 and z = 100^3/5 to equation (2)
$[ \, (100^{3/5})^x \, ]^{5/3} = 100\,000$

$(100^{3x/5})^{5/3} = 100\,000$

$100^x = 100\,000$ ‡

$10^{2x} = 10^5$

Thus,
$2x = 5$

$x = \frac{5}{2}$ → you can also use logarithm to solve for x from ‡ similar to solution for y in †.

Answer
x = 5/2
y = 5/3
z = 100^3/5

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