The slope-intercept form of a linear equation is given by $y = mx + c$, where:

• $m$ represents the slope of the line, indicating its steepness and direction.
• $c$ represents the y-intercept, the point where the line crosses the y-axis (i.e., the value of $y$ when $x=0$).

This form is incredibly useful for graphing and quickly identifying key characteristics of a line.

The point-slope form is expressed as $y - y_1 = m(x - x_1)$, where:

• $(x_1, y_1)$ is a known point on the line.
• $m$ is the slope of the line.

This form is derived directly from the definition of slope: $m = \frac{y - y_1}{x - x_1}$. It's a powerful tool for finding the equation when you don't immediately have the y-intercept.

Given two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line, the two-point form is: $\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}$

This form essentially equates the slope calculated between $(x_1, y_1)$ and any point $(x, y)$ on the line with the slope calculated between the two given points $(x_1, y_1)$ and $(x_2, y_2)$.

The intercept form of a line is $x/a + y/b = 1$, where:

• $a$ is the x-intercept (the value of $x$ when $y=0$).
• $b$ is the y-intercept (the value of $y$ when $x=0$).

This form is particularly useful when a problem provides the intercepts directly.

The general form of a linear equation is $Ax + By + C = 0$, where $A$, $B$, and $C$ are constants, and $A$ and $B$ are not both zero.

This form is highly flexible. You can convert it to slope-intercept form by solving for $y$: $y = -\frac{A}{B}x - \frac{C}{B}$ (provided $B \neq 0$). The slope is $-A/B$ and the y-intercept is $-C/B$.