To find the equation of the line that passes through the points [tex]\((-2, 6)\)[/tex] and [tex]\((1, 1)\)[/tex], follow these steps:
1. Identify coordinates of the points: The coordinates given are [tex]\( (-2, 6) \)[/tex] and [tex]\( (1, 1) \)[/tex].
2. Calculate the slope:
The slope [tex]\( m \)[/tex] of the line that goes through [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]
Substituting the coordinates:
[tex]\[
m = \frac{1 - 6}{1 - (-2)} = \frac{1 - 6}{1 + 2} = \frac{-5}{3}
\][/tex]
3. Use the point-slope form of the equation of a line:
The point-slope form of a line's equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Given one of the points [tex]\((-2, 6)\)[/tex] and the slope [tex]\( m = -\frac{5}{3} \)[/tex], plug these values into the formula:
[tex]\[
y - 6 = -\frac{5}{3}(x - (-2))
\][/tex]
Simplify the equation inside the parentheses:
[tex]\[
y - 6 = -\frac{5}{3}(x + 2)
\][/tex]
So the complete point-slope equation of the line through [tex]\((-2, 6)\)[/tex] and [tex]\((1, 1)\)[/tex] is:
[tex]\[
y - 6 = -\frac{5}{3}(x + 2)
\][/tex]
