To find the equation of the line passing through the points [tex]\((2, -2)\)[/tex] and [tex]\((4, 1)\)[/tex], we'll go through the following steps:
1. Determine the slope (m):
The slope of a line passing 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]
Plugging in the coordinates [tex]\((2, -2)\)[/tex] and [tex]\((4, 1)\)[/tex]:
[tex]\[
m = \frac{1 - (-2)}{4 - 2} = \frac{1 + 2}{4 - 2} = \frac{3}{2} = 1.5
\][/tex]
2. Find the y-intercept (b):
Using the point-slope form of the line equation, [tex]\(y - y_1 = m(x - x_1)\)[/tex], we aim to convert this into the slope-intercept form [tex]\(y = mx + b\)[/tex].
We know the slope [tex]\(m = 1.5\)[/tex] and can use either of the given points. Let's use the point [tex]\((2, -2)\)[/tex]:
[tex]\[
y - (-2) = 1.5(x - 2)
\][/tex]
Simplify the equation:
[tex]\[
y + 2 = 1.5x - 3
\][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[
y = 1.5x - 3 - 2 = 1.5x - 5
\][/tex]
So, the equation of the line in slope-intercept form is:
[tex]\[
y = 1.5x - 5
\][/tex]
Thus, the complete equation of the line through the points [tex]\((2, -2)\)[/tex] and [tex]\((4, 1)\)[/tex] is:
[tex]\[
y = 1.5x - 5
\][/tex]
