First, we need to find the slope \( m \) of the line that passes through the points \((10, 5)\) and \((4, -7)\). The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points into the formula:
[tex]\[
m = \frac{-7 - 5}{4 - 10} = \frac{-12}{-6} = 2
\][/tex]
Now, we use the point-slope form of the equation of a line, which is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
We can use either of the given points \((10, 5)\) or \((4, -7)\) in the point-slope form. Let's use \((10, 5)\):
[tex]\[
y - 5 = 2(x - 10)
\][/tex]
We check this with the given options:
A. \(y - 5 = 2(x - 10)\)
B. \(y - 5 = -2(x - 10)\)
C. \(y - 10 = 2(x - 5)\)
D. \(y - 10 = -2(x - 5)\)
The correct option that matches our point-slope equation is:
[tex]\[
\boxed{A}
\][/tex]
