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Choose the equation of the line that passes through [tex]$(5, -1)$[/tex] with the slope [tex]$-1$[/tex].

Select one or more:
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]

Sagot :

GinnyAnsweridn
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].
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