The equation of a line in slope-intercept form is given by:
\[ y = mx + b \]
where \(m\) is the slope and \(b\) is the y-intercept. In this case, the slope \(m\) is given as \(\frac{1}{4}\), and the line passes through the point \((0.2, \frac{4}{5})\).
We can substitute the slope and the coordinates of the point into the equation to find \(b\):
- Substitute \(m = \frac{1}{4}\), \(x = 0.2\), and \(y = \frac{4}{5}\):
\[ \frac{4}{5} = \frac{1}{4}(0.2) + b \]
- Calculate \(\frac{1}{4}(0.2)\):
\[ \frac{1}{4}(0.2) = \frac{0.2}{4} = \frac{0.2}{4} = 0.05 \]
- Now, substituting this value into the equation gives:
\[ \frac{4}{5} = 0.05 + b \]
- To find \(b\), we need to isolate it:
\[ b = \frac{4}{5} - 0.05 \]
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Convert \(0.05\) to a fraction to make the subtraction easier. \(0.05 = \frac{5}{100} = \frac{1}{20}\).
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Now, we need a common denominator to subtract \(\frac{4}{5}\) and \(\frac{1}{20}\). The common denominator is \(20\):
\[ \frac{4}{5} = \frac{16}{20} \]
So we have:
\[ b = \frac{16}{20} - \frac{1}{20} = \frac{16 - 1}{20} = \frac{15}{20} = \frac{3}{4} \]
Thus, the value of \(b\), the y-intercept, is:
\[ \boxed{\frac{3}{4}} \]