Reasoning: It is derived using slope $m=$ $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Reasoning: Expanding and simplifying gives $y=mx+c$.

Reasoning: Formula uses one point and slope$⟹ y-y_{1}=m(x-x_{1})$

Reasoning: Uses both points to find slope$⟹ y-y_{1}=$ $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ $\left(x-x_{1}\right)$

Reasoning: Dividing by intercepts gives standard form.

Reasoning: Shows $x$ and $y$ intercepts$⟹ \frac{x}{a}$ $+$ $\frac{y}{b}$ $=1$

Reasoning: Uses ratios of direction. Symmetric form$⟹$ $\frac{x - x_{1}}{sin θ}=\frac{y - y_{1}}{cos θ}$

Reasoning: Uses $cos α$ and $sin α$ ; Normal form $(x cos α+y sin α=p)$

Reasoning: Expanding gives $y=3x-1$ ; $y=mx+c$.

Reasoning: Directly uses intercepts.

Reasoning: Cannot represent vertical line. Slope-intercept form $y=mx+c$.

Reasoning: Uses directional components. Symmetric form$⟹$ $\frac{x - x_{1}}{sin θ}=\frac{y - y_{1}}{cos θ}$

Reasoning: $p$ is perpendicular distance ; Normal form $(x cos α+y sin α=p)$

Reasoning: One point $+$ slope$⟹$point-slope$⟹ y-y_{1}=$ $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ $\left(x-x_{1}\right)$

Reasoning: Gives slope and intercept directly$⟹$ $y=mx+c$

Reasoning: Represents intercepts.

Reasoning: Equal intercepts on both axes$⟹$ $\frac{x}{a}$ $+$ $\frac{y}{b}$ $=1$

Reasoning: Uses intercepts only$⟹$ $\frac{x}{a}$ $+$ $\frac{y}{b}$ $=1$

Reasoning: Uses direction ratios. Symmetric form$⟹$ $\frac{x - x_{1}}{sin θ}=\frac{y - y_{1}}{cos θ}$

Reasoning: Angle $α$ is with $x$-axis ; Normal form $(x cos α+y sin α=p)$

Reasoning: Expand and rearrange.

Reasoning: Vertical line has no $x$-intercept.

Reasoning: Contains $cos α$ and $sin α$.

Reasoning: Uses geometry (angle) $+$ algebra
Normal form $(x cos α+y sin α=p)$

Reasoning: They are equivalent representations.