Question

Write an equation in point-slope form for a line that passes through the following point and has the following slope:

(3,6)
Slope: -1/3
A) y - 6 = (-1/3)(x - 3)
B) y - 6 = (-3)(x - 1/3)
C) y - 6 = (1/3)(x - 3)
D) y - 6 = (3)(x - 1/3)

Answer 1

The correct equation in point-slope form for the line passing through the point (3,6) with a slope of -1/3 is A) \(y - 6 = (-1/3)(x - 3)\).

Step-by-step explanation:

To obtain the point-slope form equation, we utilize the formula
`\(y - y_1 = m(x - x_1)\),` where
`\((x_1, y_1)\)` is the given point, and
`\(m\)` is the slope. For this problem, substituting the values
`\((3,6)\) for \((x_1, y_1)\)` and
`\(-1/3\) for \(m\)`, we get
`\(y - 6 = (-1/3)(x - 3)\)`. Expanding and simplifying, this yields
`\(y - 6 = (-1/3)x + 1\)`, and further rearranging provides the correct point-slope form:
`\(y - 6 = (-1/3)(x - 3)\)`.

The slope-intercept form \(y = mx + b\) clarifies the connection between slope and y-intercept. The given slope \(-1/3\) signifies a negative slope, indicating a downward trend. By incorporating the coordinates
`\((3,6)\)`, we deduce the y-intercept as 1. Consequently, the derived equation
`\(y - 6 = (-1/3)(x - 3)\)`accurately expresses the line's behavior through the specified point with the provided slope.