# Solving a Simultaneous Equation.

I'm in no way unfamiliar with these sorts of equations and have had plenty of practice with them, however I'm going insane over a question I feel I should be able to easily answer algebraically but am stuck on. This is the question:

Solve for $x$ & $y$.

1: $2^x = y$

2: $3-x = y$

Sorry for boring you with such a mundane question, but its driving me mental.

Thanks for any help.

(Sorry if there are any formatting issues, I'm new to this StackExchange).

• Hint: have you tried to eliminate $y$? – bof Jan 8 '17 at 7:39
• Eliminating $y$ you get the equation $2^x=3-x$. – bof Jan 8 '17 at 7:41
• Putting all the $x$s on one side, $x+2^x=3$. – bof Jan 8 '17 at 7:41
• Since the function $x+2^x$ is strictly increasing, there can't be more than one solution. – bof Jan 8 '17 at 7:42
• Thanks for all the solutions! I realised how to eliminate y, I was more concerned with solving 2^x = 3−x. I never knew what a transcendental function was, so thanks to everyone for mentioning it. – Etched Jan 8 '17 at 7:52

Since both RHS = y is equivalent to the single equation. $$2^x = 3-x.$$

If you haven't found a way to solve this algebraically, I don't blame yourself. There is no way to 'solve for x' in algebra. This is a transcendental function and must be solved graphically.

If you graph $2^x$ and $3-x$, you'll see they intersect once. By inspection $x =1$ is a solution, so that's the one real solution.

(additionally there are complex solutions in terms of the Lambert W-function)

For the equation, you have to solve $2^x=3-x.$

For this kind of transcendental equation, we may not have a standard way to solve it. My hint on this is, it has an obvious solution $x=1,$ and you may be able to argue that if this equation only have one solution. Maybe by drawing the graph, and give some discussions.

We can easily see these equations are satisfied for $x=1,y=2$.

To prove that it is the only solution.

$\ \ \ \ \ 3-x=2^x$

$\Rightarrow 3=x+2^x$

$x+2^x$ is monotonically increasing function. Hence $Y=3$ intersects $Y=x+2^x$ curve at only one-point.