To determine the point-slope equation of the line passing through the points [tex]\((-5, 7)\)[/tex] and [tex]\((-4, 0)\)[/tex], we'll follow these steps:
1. Identify the coordinates of the points:
[tex]\[
(x_1, y_1) = (-5, 7)
\][/tex]
[tex]\[
(x_2, y_2) = (-4, 0)
\][/tex]
2. Calculate the slope [tex]\( m \)[/tex] of the line:
The formula for the slope between 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 coordinates:
[tex]\[
m = \frac{0 - 7}{-4 - (-5)} = \frac{-7}{-4 + 5} = \frac{-7}{1} = -7
\][/tex]
3. Use the point-slope form of the equation of a line:
The point-slope form is written as:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting [tex]\( m = -7 \)[/tex], [tex]\( x_1 = -5 \)[/tex], and [tex]\( y_1 = 7 \)[/tex]:
[tex]\[
y - 7 = -7(x + 5)
\][/tex]
Here, we adjust the expression inside the parentheses to account for [tex]\( x_1 = -5 \)[/tex]:
[tex]\[
x - (-5) = x + 5
\][/tex]
Thus, the point-slope equation of the line through [tex]\((-5, 7)\)[/tex] and [tex]\((-4, 0)\)[/tex] is:
[tex]\[
y - 7 = -7(x + 5)
\][/tex]
