Certainly! Let's write the equation of the line using the Point-Slope Form, given the slope [tex]\( m = -\frac{5}{3} \)[/tex] and the point [tex]\((-1, 5)\)[/tex].
The Point-Slope Form equation of a line is:
[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.
1. Identify the slope [tex]\( m \)[/tex]:
[tex]\[ m = -\frac{5}{3} \][/tex]
2. Identify the point [tex]\((x_1, y_1)\)[/tex]:
[tex]\[ (x_1, y_1) = (-1, 5) \][/tex]
3. Substitute [tex]\( m \)[/tex], [tex]\( x_1 \)[/tex], and [tex]\( y_1 \)[/tex] into the Point-Slope Form equation:
[tex]\[ y - 5 = -\frac{5}{3}(x - (-1)) \][/tex]
4. Simplify the expression inside the parentheses:
[tex]\[ y - 5 = -\frac{5}{3}(x + 1) \][/tex]
Therefore, the equation of the line in Point-Slope Form with a slope of [tex]\( -\frac{5}{3} \)[/tex] and passing through the point [tex]\((-1, 5)\)[/tex] is:
[tex]\[ y - 5 = -\frac{5}{3}(x + 1) \][/tex]
Now, let's match this equation with the given options:
(A) [tex]\( y-5=-3(x-1) \)[/tex] — This is incorrect.
(B) [tex]\( y+5=-\frac{5}{3}(x-3) \)[/tex] — This is incorrect.
(C) [tex]\( y-5=-\frac{5}{3}(x+1) \)[/tex] — This is correct.
(D) [tex]\( y-3=5(x-1) \)[/tex] — This is incorrect.
Thus, the correct answer is (C) [tex]\( y-5=-\frac{5}{3}(x+1) \)[/tex].
