Straight Lines - Various Forms of the Equation of a Line: Horizontal/Vertical, Point-Slope, Two-Point, Slope-Intercept, Intercept Form

Find the equation of the line passing through the point $(2, -3)$ with a slope of $4$.

1. Identify the given values: $(x_1, y_1) = (2, -3)$ and $m = 4$.
2. Use the Point-Slope form: $y - y_1 = m(x - x_1)$.
3. Substitute the values: $y - (-3) = 4(x - 2)$.
4. Simplify: $y + 3 = 4x - 8$.
5. Rearrange into general form: $4x - y - 11 = 0$.

We use the point-slope form because we are given one specific point and the gradient of the line.

Find the equation of the line that cuts off intercepts $3$ and $-2$ on the x and y axes respectively.

1. Identify the intercepts: $a = 3$ (x-intercept) and $b = -2$ (y-intercept).
2. Use the Intercept form: $\frac{x}{a} + \frac{y}{b} = 1$.
3. Substitute the values: $\frac{x}{3} + \frac{y}{-2} = 1$.
4. Find the common denominator to simplify: $\frac{2x - 3y}{6} = 1$.
5. Final equation: $2x - 3y = 6$ or $2x - 3y - 6 = 0$.

The intercept form is the most direct method here as it utilizes the points $(3, 0)$ and $(0, -2)$ where the line crosses the axes.

1. Find the slope $m$: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - (-1)}{3 - 1} = \frac{6}{2} = 3$
2. Use the Two-Point Form (or Point-Slope Form with point $A$): $y - (-1) = 3(x - 1)$ $y + 1 = 3x - 3$ $3x - y - 4 = 0$

We first calculate the slope of the line using the two given points. Then, we substitute the slope and one of the points into the point-slope equation to find the general linear equation.

1. Identify the given values: Slope $m = \frac{1}{2}$ and $y$-intercept $c = -3$.
2. Substitute into the Slope-Intercept Form $y = mx + c$: $y = \frac{1}{2}x - 3$
3. Convert to general form: $2y = x - 6$ $x - 2y - 6 = 0$

Since the slope and the $y$-intercept are directly provided, the slope-intercept form is the most efficient method to derive the equation.