To find the point-slope equation for the line that passes through the given points [tex]\((2,5)\)[/tex] and [tex]\((4,11)\)[/tex], we need to follow these steps:
1. Identify the coordinates of the points:
- The first point [tex]\(P_1\)[/tex] has coordinates [tex]\((x_1, y_1) = (2, 5)\)[/tex].
- The second point [tex]\(P_2\)[/tex] has coordinates [tex]\((x_2, y_2) = (4, 11)\)[/tex].
2. Calculate the slope [tex]\(m\)[/tex]:
Slope formula: [tex]\( m = \frac{y_2 - y_1}{x_2 - x_1} \)[/tex]
Substituting the coordinates of the points:
[tex]\[
m = \frac{11 - 5}{4 - 2} = \frac{6}{2} = 3
\][/tex]
3. Form the point-slope equation:
The point-slope form of the equation of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substitute [tex]\(m = 3\)[/tex], [tex]\(x_1 = 2\)[/tex], and [tex]\(y_1 = 5\)[/tex] into the equation:
[tex]\[
y - 5 = 3(x - 2)
\][/tex]
So the point-slope equation for the line passing through the points [tex]\((2,5)\)[/tex] and [tex]\((4,11)\)[/tex] is:
[tex]\[ y - 5 = 3(x - 2) \][/tex]
