We are given two points, A(7, -4) and B(-3, -5). We need to find: (a) The equation of the vertical line through A. (b) The equation of the horizontal line through B. (c) The slope of the line AB. (d) The slope of the line perpendicular to line AB. (e) The standard form of the equation of the line AB.

1. Problem Description

We are given two points, A(7, -4) and B(-3, -5). We need to find:

(a) The equation of the vertical line through A.

(b) The equation of the horizontal line through B.

(c) The slope of the line AB.

(d) The slope of the line perpendicular to line AB.

(e) The standard form of the equation of the line AB.

2. Solution Steps

(a) The equation of a vertical line is of the form $x = c$, where $c$ is a constant. Since the line passes through point A(7, -4), the equation of the vertical line is $x = 7$.

(b) The equation of a horizontal line is of the form $y = c$, where $c$ is a constant. Since the line passes through point B(-3, -5), the equation of the horizontal line is $y = -5$.

(c) The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Using A(7, -4) and B(-3, -5):

$m = \frac{-5 - (-4)}{-3 - 7} = \frac{-5 + 4}{-10} = \frac{-1}{-10} = \frac{1}{10}$

(d) The slope of a line perpendicular to a line with slope $m$ is $-\frac{1}{m}$.

Since the slope of line AB is $\frac{1}{10}$, the slope of a line perpendicular to AB is:

$m_{\perp} = -\frac{1}{\frac{1}{10}} = -10$

(e) The standard form of a linear equation is $Ax + By = C$.

We already know the slope of line AB is $\frac{1}{10}$. We can use the point-slope form of a line:

$y - y_1 = m(x - x_1)$

Using point A(7, -4):

$y - (-4) = \frac{1}{10}(x - 7)$

$y + 4 = \frac{1}{10}x - \frac{7}{10}$

Multiply by 10 to eliminate fractions:

$10(y + 4) = 10(\frac{1}{10}x - \frac{7}{10})$

$10y + 40 = x - 7$

Rearrange to standard form:

$-x + 10y = -47$

Multiply by -1 to make the coefficient of x positive (this is a common convention):

$x - 10y = 47$

3. Final Answer

(a) $x = 7$

(b) $y = -5$

(c) $\frac{1}{10}$

(d) $-10$

(e) $x - 10y = 47$