Sure, let's find the point-slope form of the equation for a line that has a slope of [tex]\(\frac{4}{5}\)[/tex] and passes through the point [tex]\((-2, 1)\)[/tex].
The point-slope form of the equation of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is a point on the line.
Here, the slope [tex]\( m \)[/tex] is [tex]\(\frac{4}{5}\)[/tex], and the point [tex]\((x_1, y_1)\)[/tex] is [tex]\((-2, 1)\)[/tex].
Let's substitute the given values into the point-slope form equation:
[tex]\[ y - 1 = \frac{4}{5}(x - (-2)) \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ y - 1 = \frac{4}{5}(x + 2) \][/tex]
So, the point-slope form of the line is:
[tex]\[ y - 1 = \frac{4}{5}(x + 2) \][/tex]
Now, let's compare this with the given options:
A. [tex]\( y + 1 = \frac{4}{5}(x - 2) \)[/tex]
B. [tex]\( y + 1 = \frac{4}{5}(x + 2) \)[/tex]
C. [tex]\( y - 1 = \frac{4}{5}(x + 2) \)[/tex]
D. [tex]\( y - 1 = \frac{4}{5}(x - 2) \)[/tex]
The correct equation from our calculation is [tex]\( y - 1 = \frac{4}{5}(x + 2) \)[/tex], which matches option:
C. [tex]\( y - 1 = \frac{4}{5}(x + 2) \)[/tex]
Thus, the correct answer is:
C. [tex]\( y - 1 = \frac{4}{5}(x + 2) \)[/tex]
