How can I isolate $x$ in this equation? Simultaneous equations - $7^{x-1} = 49 \hspace{1 mm} (7^{y})$ --------- (i)
$\hspace{46 mm} 3^x+3^y = 84$ ---------- (ii)
The first I make x subject of equation in this way:
$\Longrightarrow 7^{x-1} = 49 \hspace{1 mm} (7^{y})$
$\Longrightarrow 7 \times 7^x = 7^2 \times 7^y$
$\Longrightarrow 7^x = 7^{y+1}$
$\Longrightarrow x = y + 1$
The second I tried with logarithms but could not separate x in simple form:
$\Longrightarrow 3^x + 3^y = 84$
$\Longrightarrow 3^x = 84 - 3^y$
$\Longrightarrow \lg3^x = \lg (84 - 3^y)$
$\Longrightarrow x \lg3 = \lg(84 - 3^y)$
$\Longrightarrow x = \dfrac{\lg(84 - 3^y)}{\lg3}$
How can I simplify this further?
 A: You have a mistake in first equation, it should be
$$x-1 = y+2$$
from which we have $x=y+3.$ For second equation we have
$$3^{y+3}+3^y=84$$
A: First multiply equation $(i)$ by $7$ to get rid of the confusing $-1$ in the exponent and obtain $$7^x=7^{y+3}$$
Then for the second part, note that $$3^{y+3}+3^y=3^y\left(3^3+3^0\right)$$
These two key steps should make a solution rather easy.
A: You have made an error in the first equation:
$\Rightarrow 7^{x-1}=49\hspace{1 mm}(7^{y})$
$\Rightarrow \dfrac{7^{x}}{7}=49\hspace{1 mm}(7^{y})$
$\Rightarrow 7^{x}=7^{3}\hspace{1 mm}(7^{y})$
$\Rightarrow 7^{x}=7^{y+3}$
$\Rightarrow x=y+3$
Then, we can substitute this expression for $x$ into the second equation:
$\Rightarrow 3^{x}+3^{y}=84$
$\Rightarrow 3^{y+3}+3^{y}=84$
$\Rightarrow 27\cdot{3^{y}}+3^{y}=84$
$\Rightarrow 28\cdot{3^{y}}=84$
$\Rightarrow {3^{y}}=3^{1}$
$\Rightarrow y=1$
Now, let's substitute this value of $y$ into the first equation:
$\Rightarrow x=(1)+3$
$\hspace{8.75 mm}=4$
Therefore, $x=4$ and $y=1$.
A: Your solution is incorrect.
It should be as follows:
$$7^{x-1}=49\cdot 7^y$$
$$ \implies 7^{x-1}=7^{y+2}$$
Comparing both sides of the equation, we get
$$x-1=y+2$$
$$ \implies x=y+3\tag1$$
From the second equation, we have
$$3^x+3^y=84$$
$$ \implies 3^{y+3}+3^y=84$$
$$ \implies 3^y(3^3+1)=84$$
$$ \implies 3^y\cdot 28=84$$
$$ \implies 3^y=3=3^1$$
Again comparing both sides of the equation, we get
$$y=1$$
And using $(1)$, we get
$$x=4$$
Hence the required solution is
$$\boxed{x=4,y=1}$$
Hope this helps you.
