Complete the point-slope equation of the line through (1,0) and (6,-3). Use exact numbers.

[tex]\( y - (-3) = \square \)[/tex]



Answer :

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]