To find the point-slope equation for the line that passes through the points [tex]\( (7, -21) \)[/tex] and [tex]\( (-4, 23) \)[/tex], follow these steps:
1. Identify the coordinates of the given points:
- First point [tex]\( (x_1, y_1) = (7, -21) \)[/tex]
- Second point [tex]\( (x_2, y_2) = (-4, 23) \)[/tex]
2. Calculate the slope [tex]\( m \)[/tex] using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates, we get:
[tex]\[
m = \frac{23 - (-21)}{-4 - 7}
\][/tex]
Simplifying the expression:
[tex]\[
m = \frac{23 + 21}{-4 - 7} = \frac{44}{-11} = -4
\][/tex]
So, the slope [tex]\( m \)[/tex] is [tex]\( -4 \)[/tex].
3. Use the point-slope form of the equation of a line:
The point-slope form is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Using the first point [tex]\( (7, -21) \)[/tex]:
[tex]\[
y - (-21) = -4(x - 7)
\][/tex]
4. Simplify the equation:
[tex]\[
y + 21 = -4(x - 7)
\][/tex]
Thus, the point-slope equation for the line that passes through the points [tex]\( (7, -21) \)[/tex] and [tex]\( (-4, 23) \)[/tex] is:
[tex]\[
y + 21 = -4(x - 7)
\][/tex]
In this equation, [tex]\( y - [-21] = -4 (x - 7) \)[/tex]. Therefore, the blanks are filled as follows:
[tex]\[
y - [-21] = -4 (x - 7)
\][/tex]
So, the final point-slope form of the equation is:
[tex]\[
y - (-21) = -4 (x - 7)
\][/tex]
Or more simply:
[tex]\[
\boxed{y + 21 = -4 (x - 7)}
\][/tex]
