To find the slope of the line passing through the points [tex]\(\left(\frac{5}{8}, -\frac{1}{2}\right)\)[/tex] and [tex]\(\left(\frac{1}{4}, \frac{1}{4}\right)\)[/tex], we can follow these steps:
1. Identify the coordinates of the two points:
- Point 1: [tex]\(\left(\frac{5}{8}, -\frac{1}{2}\right)\)[/tex]
- Point 2: [tex]\(\left(\frac{1}{4}, \frac{1}{4}\right)\)[/tex]
2. Use the slope formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] are the coordinates of the first point and [tex]\((x_2, y_2)\)[/tex] are the coordinates of the second point.
3. Substitute the coordinates into the formula:
[tex]\[
m = \frac{\frac{1}{4} - \left(-\frac{1}{2}\right)}{\frac{1}{4} - \frac{5}{8}}
\][/tex]
4. Calculate the difference in the y-coordinates:
[tex]\[
y_2 - y_1 = \frac{1}{4} - \left(-\frac{1}{2}\right)
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
y_2 - y_1 = \frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4}
\][/tex]
So, the difference in the y-coordinates is [tex]\(\frac{3}{4}\)[/tex].
5. Calculate the difference in the x-coordinates:
[tex]\[
x_2 - x_1 = \frac{1}{4} - \frac{5}{8}
\][/tex]
To subtract these fractions, convert [tex]\(\frac{1}{4}\)[/tex] to a fraction with a common denominator:
[tex]\[
\frac{1}{4} = \frac{2}{8}
\][/tex]
So:
[tex]\[
x_2 - x_1 = \frac{2}{8} - \frac{5}{8} = \frac{2 - 5}{8} = \frac{-3}{8}
\][/tex]
Therefore, the difference in the x-coordinates is [tex]\(-\frac{3}{8}\)[/tex].
6. Divide the difference in the y-coordinates by the difference in the x-coordinates:
[tex]\[
m = \frac{\frac{3}{4}}{-\frac{3}{8}}
\][/tex]
7. Simplify the division of fractions:
[tex]\[
m = \frac{3}{4} \times \frac{8}{-3}
\][/tex]
Multiply the fractions:
[tex]\[
m = \frac{3 \cdot 8}{4 \cdot (-3)} = \frac{24}{-12} = -2
\][/tex]
Thus, the slope of the line passing through the points [tex]\(\left(\frac{5}{8}, -\frac{1}{2}\right)\)[/tex] and [tex]\(\left(\frac{1}{4}, \frac{1}{4}\right)\)[/tex] is [tex]\(-2\)[/tex].
