The straight line $y=\frac{7}{15}x + \frac{1}{3}$ passes through two points with integral coordinates $A(10, 5), B(-20, -9)$. Are there other integral points on this line?
From "Functions and Graphs", Gelfand
I found another answer here that is concerned with the number of integral points on a line:
Write the slope $\frac{y_2 - y_1}{x_2 - x_1}$ in lowest terms as $\frac{a}{b}$. Then you can obtain every point with integer coordinates by starting at $(x_1, y_1)$ and repeatedly going left $b$ units and up $a$ units. So how many points does that give you?
We already have the slope $\frac{a}{b} = \frac{7}{15}$
But I don't get why moving on the x-axis by $b$ units and on the y-axis by $a$ units necessarily will give me integer coordinates.
Because the slope is $\dfrac{\text{rise}}{\text{run}}=\dfrac{7}{15}$, you will remain on the line if you go left $15$ units and up $7$ units, starting from a point on the line. Moreover, if you start at a point with integer coordinates, adding a multiple of $15$ to its $x$ coordinate and a multiple of $7$ to its $y$ coordinate will give you integer coordinates.