Question

Find and interpret the slope of the line containing the given points.  
(1/4, 5/6) and (-11/4, -1/6)

Answer 1

To find the slope of the line containing the points \((1/4, 5/6)\) and \((-11/4, -1/6)\), we use the formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\):

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Here, we can designate:
- \(x_1 = \frac{1}{4}\), \(y_1 = \frac{5}{6}\)
- \(x_2 = -\frac{11}{4}\), \(y_2 = -\frac{1}{6}\)

Now, we calculate \(y_2 - y_1\) and \(x_2 - x_1\):

1. Calculate \(y_2 - y_1\):
\[
y_2 - y_1 = -\frac{1}{6} - \frac{5}{6} = -\frac{1 + 5}{6} = -\frac{6}{6} = -1
\]

2. Calculate \(x_2 - x_1\):
\[
x_2 - x_1 = -\frac{11}{4} - \frac{1}{4} = -\frac{11 + 1}{4} = -\frac{12}{4} = -3
\]

Now, substitute these values into the slope formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1}{-3} = \frac{1}{3}
\]

The slope of the line containing the points \((\frac{1}{4}, \frac{5}{6})\) and \((- \frac{11}{4}, - \frac{1}{6})\) is \(\frac{1}{3}\).

### Interpretation of the slope:
The slope \(\frac{1}{3}\) indicates that for every increase of 3 units in the x-direction, the y-value increases by 1 unit. This signifies a positive relationship between the x and y coordinates of the points on this line, suggesting that as one variable increases, the other does as well.