To determine the equation of the line passing through the points [tex]\((5, -3)\)[/tex] and [tex]\((-10, 15)\)[/tex], 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]\((5, -3)\)[/tex] and [tex]\((-10, 15)\)[/tex]:
[tex]\[
m = \frac{15 - (-3)}{-10 - 5} = \frac{15 + 3}{-10 - 5} = \frac{18}{-15} = -\frac{6}{5}
\][/tex]
2. Use the point-slope form to determine the equation of the line:
The point-slope form of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Using one of the points, for example, [tex]\((5, -3)\)[/tex]:
[tex]\[
y - (-3) = -\frac{6}{5}(x - 5)
\][/tex]
Simplify this equation:
[tex]\[
y + 3 = -\frac{6}{5}x + \frac{6}{5} \times 5
\][/tex]
[tex]\[
y + 3 = -\frac{6}{5}x + 6
\][/tex]
To express this in the [tex]\(y = mx + b\)[/tex] form, isolate [tex]\(y\)[/tex]:
[tex]\[
y = -\frac{6}{5}x + 6 - 3
\][/tex]
[tex]\[
y = -\frac{6}{5}x + 3
\][/tex]
Therefore, the equation of the line passing through the given points is:
[tex]\[
y = -\frac{6}{5}x + 3
\][/tex]
From the choices provided, the correct answer is:
[tex]\[
\boxed{y = -\frac{6}{5} x + 3}
\][/tex]
