How do you find the inverse of # x - 2y = 5# and is it a function?

Question

How do you find the inverse of  # x - 2y = 5# and is it a function?

Theodore Amidon · Accepted Answer

#color(blue)(f^-1(x)=2x+5)#

#x-2y=5#

#y=1/2x-5/2#

To find the inverse of this we express #x# as a function of #y#:

#x-2y=5#

#x=2y+5#

Substituting:

#y=x#

#y=2x+5#

#color(blue)(f^-1(x)=2x+5)#

Yes this is a function.

For every input there is only one output.

The inverse of a function is always the function reflected in the line #y=x#

The graph below illustrates this:

Sadie Ackerman · Answer

undefined To find the inverse of the equation $x - 2y = 5$, follow these steps:

1. Solve the equation for $y$ to express it in terms of $x$.
2. Swap $x$ and $y$ in the resulting equation.
3. Solve the new equation for $y$.

Starting with $x - 2y = 5$:
1. Solve for $y$:
$$ x - 2y = 5 $$
$$ -2y = -x + 5 $$
$$ y = \frac{-x + 5}{-2} $$
$$ y = \frac{x - 5}{2} $$

2. Swap $x$ and $y$:
$$ x = \frac{y - 5}{2} $$

3. Solve for $y$:
$$ 2x = y - 5 $$
$$ y = 2x + 5 $$

So, the inverse of the equation $x - 2y = 5$ is $y = 2x + 5$.

As for whether it represents a function, yes, it does. Both the original equation and its inverse represent functions because for each value of $x$, there corresponds exactly one value of $y$ and vice versa. Therefore, both equations represent functions.