To find the equation of a line in standard form [tex]\(Ax + By = C\)[/tex] that passes through the points [tex]\((3.2, -6)\)[/tex] and [tex]\((5.2, -5)\)[/tex], let's follow these steps:
1. Find the slope (m):
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Using the given points [tex]\((x_1, y_1) = (3.2, -6)\)[/tex] and [tex]\((x_2, y_2) = (5.2, -5)\)[/tex],
[tex]\[
m = \frac{-5 - (-6)}{5.2 - 3.2} = \frac{1}{2} = 0.5
\][/tex]
2. Find the y-intercept (b):
Using the slope-intercept form [tex]\(y = mx + b\)[/tex],
[tex]\[
y = 0.5x + b
\][/tex]
Substitute one of the points, for instance, [tex]\((3.2, -6)\)[/tex], to solve for [tex]\(b\)[/tex]:
[tex]\[
-6 = 0.5(3.2) + b \implies -6 = 1.6 + b \implies b = -6 - 1.6 = -7.6
\][/tex]
So, the equation in slope-intercept form is:
[tex]\[
y = 0.5x - 7.6
\][/tex]
3. Convert to standard form [tex]\(Ax + By = C\)[/tex]:
Rewrite the equation [tex]\(y = 0.5x - 7.6\)[/tex] to standard form.
[tex]\[
y = 0.5x - 7.6
\][/tex]
Subtract [tex]\(0.5x\)[/tex] from both sides to get:
[tex]\[
-0.5x + y = -7.6
\][/tex]
Multiply through by -1 to make the coefficients more conventional:
[tex]\[
0.5x - y = 7.6
\][/tex]
But we're given the standard forms in the problem as potential answers, and it matches with:
[tex]\[
-0.5x + y = -7.6
\][/tex]
So, the correct choice among the options is:
[tex]\[
-0.5 x + y = -7.6
\][/tex]
Therefore, the equation in standard form for the line passing through the points [tex]\((3.2, -6)\)[/tex] and [tex]\((5.2, -5)\)[/tex] is:
[tex]\[
\boxed{-0.5 x + y = -7.6}
\][/tex]
