To find the point-slope form of the equation of the line passing through the points [tex]\((1,0)\)[/tex] and [tex]\((6,-3)\)[/tex], we must follow a series of steps. Let's go through them:
1. Identify the coordinates:
- Point 1: [tex]\((x_1, y_1) = (1, 0)\)[/tex]
- Point 2: [tex]\((x_2, y_2) = (6, -3)\)[/tex]
2. 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 the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates:
[tex]\[
m = \frac{-3 - 0}{6 - 1} = \frac{-3}{5} = -0.6
\][/tex]
3. Use the point-slope form of the equation:
The point-slope form of a line is given by:
[tex]\[
y - y_1 = m (x - x_1)
\][/tex]
We can use either point to write the equation. Let's use the point [tex]\((6, -3)\)[/tex]:
[tex]\[
y - (-3) = m (x - 6)
\][/tex]
4. Substitute the slope [tex]\(m\)[/tex] in the equation:
We already calculated that [tex]\(m = -0.6\)[/tex]. Using this value:
[tex]\[
y - (-3) = -0.6 (x - 6)
\][/tex]
So, the point-slope equation of the line is:
[tex]\[
y - (-3) = -0.6 (x - 6)
\][/tex]
or
[tex]\[
y + 3 = -0.6 (x - 6)
\][/tex]
Thus, the completed point-slope equation is:
[tex]\[
y - (-3) = -0.6 (x - 6)
\][/tex]
