Here are the solutions to the problems in your image:

Problem 1: Find the slope and angle of inclination of the line joining $\left( \frac{8}{3}, \frac{1}{3} \right)$ and $\left( \frac{5}{4}, \frac{13}{4} \right)$.

Solution:

The slope formula is: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the given points: $m = \frac{\frac{13}{4} - \frac{1}{3}}{\frac{5}{4} - \frac{8}{3}}$

1. Simplify the numerator: $\frac{13}{4} - \frac{1}{3} = \frac{39}{12} - \frac{4}{12} = \frac{35}{12}$

2. Simplify the denominator: $\frac{5}{4} - \frac{8}{3} = \frac{15}{12} - \frac{32}{12} = -\frac{17}{12}$

Thus, the slope is: $m = \frac{\frac{35}{12}}{-\frac{17}{12}} = \frac{35}{-17} = -\frac{35}{17}$

Angle of inclination ($\theta$):

The angle is given by: $\theta = \tan^{-1}(m) = \tan^{-1}(-\frac{35}{17})$

Problem 2: Determine $x$ so that the three points $(x, -3), (9, 1), (13, 2)$ lie on a straight line.

Solution:

If three points lie on the same line, the slopes between consecutive points must be equal. Let the points be $A(x, -3)$, $B(9, 1)$, and $C(13, 2)$.

1. Slope of $AB$: $m_{AB} = \frac{1 - (-3)}{9 - x} = \frac{4}{9 - x}$

2. Slope of $BC$: $m_{BC} = \frac{2 - 1}{13 - 9} = \frac{1}{4}$

Set $m_{AB} = m_{BC}$: $\frac{4}{9 - x} = \frac{1}{4}$

Cross-multiply: $4(9 - x) = 4 \implies 36 - 4x = 4 \implies 4x = 32 \implies x = 8$

Thus, $x = 8$.

Problem 3: A line with slope $\frac{1}{5}$ passes through $(6, 3)$. If the abscissa of a point on the line is $11$, what is its ordinate?

Solution:

The equation of the line in point-slope form is: $y - y_1 = m(x - x_1)$

Substitute $m = \frac{1}{5}$, $(x_1, y_1) = (6, 3)$: $y - 3 = \frac{1}{5}(x - 6)$

Simplify: $y - 3 = \frac{1}{5}x - \frac{6}{5}$ $y = \frac{1}{5}x + \frac{9}{5}$

Substitute $x = 11$ (abscissa): $y = \frac{1}{5}(11) + \frac{9}{5} = \frac{11}{5} + \frac{9}{5} = \frac{20}{5} = 4$

Thus, the ordinate is $4$.

Any questions, or should I provide further details on any solution?

Here are five related questions to expand your understanding:

1. What does the slope of a line represent geometrically?
2. How can you find the equation of a line given two points?
3. How do you verify if three points are collinear using determinants?
4. What is the significance of the angle of inclination of a line?
5. How can point-slope and slope-intercept forms of a line be converted into each other?

Tip: Always double-check arithmetic in slope calculations, as small errors propagate through!