To find the equation of the line through the points [tex]\((-5, 4)\)[/tex] and [tex]\((1, 6)\)[/tex] using the point-slope form, follow these steps:
1. Calculate the Slope:
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 the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
2. Plug in the Coordinates:
[tex]\[
x_1 = -5, \quad y_1 = 4, \quad x_2 = 1, \quad y_2 = 6
\][/tex]
Calculate the slope [tex]\(m\)[/tex]:
[tex]\[
m = \frac{6 - 4}{1 - (-5)} = \frac{2}{1 + 5} = \frac{2}{6} = \frac{1}{3}
\][/tex]
3. Use the Point-Slope Form:
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Choose the point [tex]\((1, 6)\)[/tex]:
Plug in the slope [tex]\(m\)[/tex] and the point [tex]\((1, 6)\)[/tex] into the point-slope form:
[tex]\[
y - 6 = \frac{1}{3}(x - 1)
\][/tex]
Hence, the complete point-slope equation of the line through [tex]\((-5, 4)\)[/tex] and [tex]\((1, 6)\)[/tex] is:
[tex]\[
y - 6 = \frac{1}{3}(x - 1)
\][/tex]
So, filling in the blank, we get:
[tex]\[
y - 6 = \frac{1}{3}(x - 1)
\][/tex]
