To find the equation of the line passing through the points [tex]\((5, 21)\)[/tex] and [tex]\((-5, -29)\)[/tex], we can follow these steps:
### Step 1: Identify the coordinates of the points
The coordinates of the first point are [tex]\((x_1, y_1) = (5, 21)\)[/tex], and the coordinates of the second point are [tex]\((x_2, y_2) = (-5, -29)\)[/tex].
### Step 2: Calculate the slope [tex]\(m\)[/tex]
The formula for the slope [tex]\(m\)[/tex] of the line passing through 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 coordinates:
[tex]\[
m = \frac{-29 - 21}{-5 - 5} = \frac{-50}{-10} = 5
\][/tex]
So, the slope [tex]\(m\)[/tex] is [tex]\(5.0\)[/tex].
### Step 3: Find the y-intercept [tex]\(b\)[/tex]
The equation of a line in slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
We can find the y-intercept [tex]\(b\)[/tex] by using one of the points and the calculated slope. Let's use the point [tex]\((5, 21)\)[/tex]:
[tex]\[
21 = 5 \cdot 5 + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
21 = 25 + b
\][/tex]
[tex]\[
b = 21 - 25
\][/tex]
[tex]\[
b = -4
\][/tex]
The y-intercept [tex]\(b\)[/tex] is [tex]\(-4.0\)[/tex].
### Step 4: Write the equation of the line
Now that we have the slope [tex]\(m = 5.0\)[/tex] and the y-intercept [tex]\(b = -4.0\)[/tex], the equation of the line is:
[tex]\[
y = 5x - 4
\][/tex]
So, the equation of the line passing through the points [tex]\((5, 21)\)[/tex] and [tex]\((-5, -29)\)[/tex] is:
[tex]\[
y = 5.0x - 4.0
\][/tex]
