To find the equation of the line that passes through the points \((0, \frac{3}{5})\) and \((- \frac{1}{2}, \frac{3}{4})\), we will first find the slope \(m\) of the line:
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Plugging in the given points \((0, \frac{3}{5})\) and \((- \frac{1}{2}, \frac{3}{4})\):
- \(x_1 = 0\), \(y_1 = \frac{3}{5}\)
- \(x_2 = -\frac{1}{2}\), \(y_2 = \frac{3}{4}\)
Now we calculate the slope:
\[ m = \frac{\frac{3}{4} - \frac{3}{5}}{-\frac{1}{2} - 0} \]
To do \(\frac{3}{4} - \frac{3}{5}\), we need a common denominator. The least common multiple of 4 and 5 is 20. So, we can convert the fractions:
\[ \frac{3}{4} = \frac{15}{20}, \quad \frac{3}{5} = \frac{12}{20} \]
Thus,
\[ \frac{3}{4} - \frac{3}{5} = \frac{15}{20} - \frac{12}{20} = \frac{3}{20} \]
Next, substituting back into our slope equation gives:
\[ m = \frac{\frac{3}{20}}{-\frac{1}{2}} = \frac{3}{20} \cdot \left(-2\right) = -\frac{3}{10} \]
Now we have the slope \(m = -\frac{3}{10}\).
Next, we can use the point-slope form of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
Using the point \((0, \frac{3}{5})\):
\[ y - \frac{3}{5} = -\frac{3}{10}(x - 0) \]
This simplifies to:
\[ y - \frac{3}{5} = -\frac{3}{10}x \]
Adding \(\frac{3}{5}\) to both sides:
\[ y = -\frac{3}{10}x + \frac{3}{5} \]
Now, we can express \(\frac{3}{5}\) with a denominator of 10:
\[ \frac{3}{5} = \frac{6}{10} \]
So, the final equation of the line is:
\[ y = -\frac{3}{10}x + \frac{6}{10} \]
Thus, the correct response among the options given is:
y = -\frac{3}{10}x + \frac{3}{5} which is equivalent to y = -\frac{3}{10}x + 0.6.