Certainly! To find the equation of the line passing through the points [tex]\((1, -5)\)[/tex] and [tex]\((9, 11)\)[/tex], we'll follow these steps:
1. Determine the slope [tex]\(m\)[/tex] of the line:
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points [tex]\((x_1, y_1) = (1, -5)\)[/tex] and [tex]\((x_2, y_2) = (9, 11)\)[/tex]:
[tex]\[
m = \frac{11 - (-5)}{9 - 1} = \frac{11 + 5}{9 - 1} = \frac{16}{8} = 2
\][/tex]
2. Find the y-intercept [tex]\(b\)[/tex]:
The equation of a line in slope-intercept form is [tex]\(y = mx + b\)[/tex]. We can solve for [tex]\(b\)[/tex] by using one of the points and the value of the slope [tex]\(m\)[/tex]:
[tex]\[
y = mx + b
\][/tex]
Using the point [tex]\((1, -5)\)[/tex] and the slope [tex]\(m = 2\)[/tex]:
[tex]\[
-5 = 2(1) + b \implies -5 = 2 + b \implies b = -5 - 2 = -7
\][/tex]
3. Write the equation of the line:
Now that we have both the slope [tex]\(m\)[/tex] and the y-intercept [tex]\(b\)[/tex], we can write the equation of the line:
[tex]\[
y = 2x - 7
\][/tex]
Therefore, the equation of the line passing through the points [tex]\((1, -5)\)[/tex] and [tex]\((9, 11)\)[/tex] is:
[tex]\[
y = 2x - 7
\][/tex]
