Pearson Correlation Coefficient for samples $X,Y$ equals $1$ iff there is $a>0, b\in \mathbb{R}$ with $x_i = ay_i + b$ for every $i=1,\dots,n$.

Question

Let $X = (x_1,\dots,x_n)$ and $Y = (y_1,\dots,y_n)$ be outcome values. We have defined the variances $s^2_X = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$, $s^2_Y = \frac{1}{n-1} \sum_{i=1}^n (y_i - \bar{y})^2$, the covariance $s_{XY} = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})$ and the correlation coefficient $r_{XY} = \dfrac{s_{XY}}{s_X s_Y}$.

The if-direction was rather easy to show but I have trouble with the other direction. I do not know how to find $a>0, b \in \mathbb{R}$ or how to use my precondition $r_{XY}=1$ or how to show the equality for every sample unit.

Could anyone help me with this problem? Thank you.