To determine the equation of the line that passes through the point [tex]\((5, -1)\)[/tex] and has a slope of [tex]\(-1\)[/tex], we can use the point-slope form of the linear equation. The point-slope form is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m\)[/tex] is the slope.
Given:
- The point [tex]\((x_1, y_1) = (5, -1)\)[/tex]
- The slope [tex]\(m = -1\)[/tex]
Substituting these values into the point-slope form, we get:
[tex]\[ y - (-1) = -1(x - 5) \][/tex]
Simplifying this, we get:
[tex]\[ y + 1 = -1(x - 5) \][/tex]
So, the equation of the line is:
[tex]\[ y + 1 = -1(x - 5) \][/tex]
Now, let's compare this with the given options:
a. [tex]\( y - 1 = (x + 5) \)[/tex]
b. [tex]\( y - 5 = (x + 1) \)[/tex]
c. [tex]\( x + y = 4 \)[/tex]
d. [tex]\( y + 1 = -(x - 5) \)[/tex]
The correct option that matches the derived equation [tex]\( y + 1 = -1(x - 5) \)[/tex] is:
d. [tex]\( y + 1 = -(x - 5) \)[/tex]
Thus, the equation of the line that passes through [tex]\((5, -1)\)[/tex] with a slope of [tex]\(-1\)[/tex] is [tex]\( y + 1 = -(x - 5) \)[/tex]. The correct option is [tex]\(d\)[/tex].
