A line through (3, 7) makes an angle of 45° with the positive x-axis…

Through $(x_1, y_1)$ with slope $m$: $y - y_1 = m(x - x_1)$. Slope and $y$-intercept $c$: $y = mx + c$. Intercepts $a$ and $b$: $\dfrac{x}{a} + \dfrac{y}{b} = 1$. The point dividing $AB$ in the ratio $m : n$ is $\left(\dfrac{m x_2 + n x_1}{m + n}, \dfrac{m y_2 + n y_1}{m + n}\right)$.