To find the linear equation between the points [tex]\((-1, 5)\)[/tex] and [tex]\((3, 2)\)[/tex], we follow these steps:
1. Calculate the Slope (m):
The slope of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points:
[tex]\[
m = \frac{2 - 5}{3 - (-1)} = \frac{2 - 5}{3 + 1} = \frac{-3}{4} = -0.75
\][/tex]
2. Calculate the y-intercept (b):
The equation of a line in slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
To find [tex]\(b\)[/tex], we will use one of the given points, say [tex]\((-1, 5)\)[/tex]. Substituting [tex]\(x = -1\)[/tex] and [tex]\(y = 5\)[/tex] into the equation, along with [tex]\(m = -0.75\)[/tex]:
[tex]\[
5 = (-0.75)(-1) + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
5 = 0.75 + b
\][/tex]
[tex]\[
b = 5 - 0.75 = 4.25
\][/tex]
3. Form the Equation:
Now that we have the slope ([tex]\(m = -0.75\)[/tex]) and the y-intercept ([tex]\(b = 4.25\)[/tex]), the equation of the line is:
[tex]\[
y = -0.75x + 4.25
\][/tex]
Alternatively, writing it in fraction form:
[tex]\[
y = -\frac{3}{4}x + \frac{17}{4}
\][/tex]
Thus, the correct equation of the line is:
d. [tex]\(y = -\frac{3}{4}x + \frac{17}{4}\)[/tex]
So the correct answer is [tex]\( \boxed{d} \)[/tex].
