To determine the equation of the line that passes through the points [tex]\((8, -1)\)[/tex] and [tex]\((2, -5)\)[/tex] and express it in standard form, let's follow these steps:
1. Calculate the Slope (m):
The slope [tex]\(m\)[/tex] 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]\((8, -1)\)[/tex] and [tex]\((2, -5)\)[/tex]:
[tex]\[
m = \frac{-5 - (-1)}{2 - 8} = \frac{-5 + 1}{2 - 8} = \frac{-4}{-6} = \frac{2}{3}
\][/tex]
2. Point-Slope Form of the Line:
Using the point-slope form [tex]\(y - y_1 = m(x - x_1)\)[/tex] with [tex]\(m = \frac{2}{3}\)[/tex], [tex]\(x_1 = 8\)[/tex], and [tex]\(y_1 = -1\)[/tex]:
[tex]\[
y + 1 = \frac{2}{3}(x - 8)
\][/tex]
3. Convert to Standard Form:
Standard form of a line is [tex]\(Ax + By = C\)[/tex], where [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] are integers, and [tex]\(A \geq 0\)[/tex].
Start by eliminating the fraction by multiplying every term by 3:
[tex]\[
3(y + 1) = 2(x - 8)
\][/tex]
Distribute on both sides:
[tex]\[
3y + 3 = 2x - 16
\][/tex]
Rearrange to get the equation in the form [tex]\(Ax + By = C\)[/tex]:
[tex]\[
2x - 3y = 19
\][/tex]
Therefore, the equation of the line in standard form is:
[tex]\[
\boxed{2x - 3y = 19}
\][/tex]
The steps have been followed to ensure all calculations are shown, justifying this result.
