Post a Comment
In my previous post, we discussed simultaneous equations, today, we will solidify our knowledge on simulteanous equations by looking at the exponential part of the simulteanous equation.

A simultaneous exponential equation is simply a simultaneous equation derived from an exponent.

For example, in the  following equation

The accompanying simulteanous equations can be derived

Where the value of $x$ and $y$ that satisfy this equation is $2$ and $1$ respectively

But, How do we get the answer?
The accompanying example illustrates the steps to solving a simulteanous exponential equation.

👉 This post is a continuation of our series on indices 👈

Example 1
Find the value of $x$ and $y$ given that $4^{x+1}=64^{y-1}$, $(\frac{1}{4})^{2x}=(\frac{1}{16})^{y+2}$, 

First, you rewrite all the number in base 4
The base is the same, hence, we equate the power
$x-3y=-4$........eqn 1


The base is the same, we equate the power.
$-2x+2y=4$........eqn 2

Let's make x the subject of the formula in Eqn 1.


Substituting the value of $x$ in Eqn 2

Substituting y in eqn $x=-4+3y$

In the above example, we solve this simultaneous using the substitution method, if you're having not yet familiar with the simultaneous equation by substitution, see this post

Example 2
If $9^{x-1}=81^{1-y}$, $25^x=5^{3-y}$, find $x$ and $y$

Let's rewrite each equation so that both sides of each equation has the same base.


The base is the same, we equate the power.

$x+2y=3$......eqn 1

Let's do the same for the other eqn

Equating the power
$2x+y=3$.....eqn 2

Making $x$ the subject of the formula In equation 1


Substituting $x$ in eqn 2

Divide both side by $3$

Substituting the value if $y$ in $x=3-2y$

Example 3
Solve for $x$ and $y$ in the equation $3^{x-y}=27$, $3^{3-y}=2187$


Equating the powder
$x-y=3$...... eqn 1


Substituting the value of y in Eqn 1


Example 4
Solve $9^{1-x}=27^y$, $x-y=\frac{-3}{2}$

Expressing $9^{1-x}=27^y$ in the base of 3

As usual, we equate the exponent
$-2x-3y=-2$......eqn 1

$x-y=\frac{-3}{2}$....eqn 2

Let make $x$ the subject of the formula in eqn 2

Substituting the value of $x$ in eqn 1

Divide both side by $-5$

Substituting $y$ in $x=\frac{-3+2y}{2}$

Related posts

Example 5
Solve the simulteanous exponential equation $3^x-2^{y+2}=49$, $2^y-3^{x-2}=-1$

Unlike the previous one, here, the base is not the same, so, let's try to make the two equations look similar.

This can be rewritten as
$3^x-2^y(4)=49$.....eqn 1

This can be rewritten as
$2^y-\frac{3^x}{9}=-1$.....eqn 2

Let $2^y=k$, $3^x=l$
$l-4k=49$......eqn 3


Taking the L.C.M of both side

Cross multiply 
$-l+9k=-9$....eqn 4

We are going to be solving this equation by elimination. Add eqn 3 and 4


Divide both side by 5

Substituting k in eqn 3

Recalled that $2^y=k$, $3^x=l$


Finally, we ended our series on indices. I believe you should be able to solve any question on indices. But still, if you have got a question, do well to ask our telegram community.
Help us grow our readership by sharing this post

Related Posts

Post a Comment

Subscribe Our Newsletter