To find the equation of a line given its slope and a point through which it passes, we can use the slope-intercept form of a line, which is:
[tex]\[ y = mx + b \][/tex]
Here, [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
Given:
- The slope [tex]\( m \)[/tex] is [tex]\( \frac{-1}{3} \)[/tex].
- The point through which the line passes is [tex]\( \left(0, \frac{7}{9}\right) \)[/tex].
Let's follow the steps to find the equation of the line:
1. Insert the Given Values:
Insert the point [tex]\((0, \frac{7}{9})\)[/tex] and the slope [tex]\( m = \frac{-1}{3} \)[/tex] into the equation of the line.
So, we have:
[tex]\[ y = mx + b \][/tex]
2. Substitute the Point into the Equation:
Since the x-coordinate of the point is 0, substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = \frac{7}{9} \)[/tex] into the equation:
[tex]\[ \frac{7}{9} = \left(\frac{-1}{3}\right)(0) + b \][/tex]
3. Simplify the Equation:
Simplifying the above equation, we get:
[tex]\[ \frac{7}{9} = 0 + b \][/tex]
Therefore:
[tex]\[ b = \frac{7}{9} \][/tex]
4. Write the Equation of the Line:
Now, we have determined that the y-intercept [tex]\( b \)[/tex] is [tex]\( \frac{7}{9} \)[/tex] and the slope [tex]\( m \)[/tex] is [tex]\( \frac{-1}{3} \)[/tex]. Substituting these values into the slope-intercept form, the equation of the line is:
[tex]\[ y = \left(\frac{-1}{3}\right)x + \frac{7}{9} \][/tex]
So, the final equation of the line with a slope of [tex]\( \frac{-1}{3} \)[/tex] that passes through the point [tex]\((0, \frac{7}{9})\)[/tex] is:
[tex]\[ y = \left(\frac{-1}{3}\right)x + \frac{7}{9} \][/tex]
