If the line y = 2x+3 is reflected in the line y = x+1, find what its equation becomes.
When you reflect a line over $\displaystyle y = x$, you simply switch the x and y variables.
So $\displaystyle y = 2x + 3$ becomes
$\displaystyle x = 2y + 3$
$\displaystyle 2y = x - 3$
$\displaystyle y = \frac{1}{2}x - \frac{3}{2}$
The resulting equation of a line is also known as the inverse of the original equation.
In the problem, the line was reflected over $\displaystyle y = x + 1$, so you add $\displaystyle 1$ to the inverse of the original equation.
$\displaystyle y = \frac{1}{2}x - \frac{3}{2} + 1$
$\displaystyle y = \frac{1}{2}x - \frac{1}{2}$
If you drew $\displaystyle y = 2x+3$ and $\displaystyle y = \frac{1}{2}x - \frac{1}{2}$ on graph paper and folded the paper at $\displaystyle y = x + 1$, the two lines should be right on top of each other.