To determine the equation of the line passing through the points [tex]\((-2, 9)\)[/tex] and [tex]\((-9, 2)\)[/tex], we follow these steps:
1. Calculate the slope [tex]\(m\)[/tex]:
The slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points [tex]\((-2, 9)\)[/tex] and [tex]\((-9, 2)\)[/tex] into the formula:
[tex]\[
m = \frac{2 - 9}{-9 - (-2)} = \frac{2 - 9}{-9 + 2} = \frac{-7}{-7} = 1
\][/tex]
2. Find the y-intercept [tex]\(b\)[/tex]:
The equation of the line in slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
We already have the slope [tex]\(m = 1\)[/tex]. Now, we need to find the y-intercept [tex]\(b\)[/tex]. We can use one of the given points and substitute [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(m\)[/tex] to solve for [tex]\(b\)[/tex]. We use point [tex]\((-2, 9)\)[/tex]:
[tex]\[
9 = 1 \cdot (-2) + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
9 = -2 + b \implies b = 9 + 2 = 11
\][/tex]
3. Form the equation of the line:
Substituting the values of [tex]\(m\)[/tex] and [tex]\(b\)[/tex] into the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
y = 1x + 11
\][/tex]
This simplifies to:
[tex]\[
y = x + 11
\][/tex]
Therefore, the equation of the line passing through the points [tex]\((-2, 9)\)[/tex] and [tex]\((-9, 2)\)[/tex] is:
[tex]\[
\boxed{y = x + 11}
\][/tex]
