Graph the line passing through the given point and having the indicated slope. Plot two points on the line. through (3, -4), m = - 1/3

Verified step by step guidance

1. Identify the given point and slope. The point is (3, -4) and the slope is \( m = -\frac{1}{3} \).
2. Use the point-slope form of the equation of a line: \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the given point.
3. Substitute the given point and slope into the point-slope form: \( y - (-4) = -\frac{1}{3}(x - 3) \). Simplify this to get \( y + 4 = -\frac{1}{3}(x - 3) \).
4. Find the y-coordinate of a second point by choosing a value for \( x \) different from 3, substitute it into the equation, and solve for \( y \). This gives you a second point on the line.
5. Use the slope \( m = -\frac{1}{3} \) to find a third point by moving 3 units horizontally and 1 unit vertically from the original point, following the slope direction. Plot the original point and these two points, then draw the line through them.

Key Concepts

Slope of a Line

The slope represents the rate of change or steepness of a line, calculated as the ratio of the vertical change to the horizontal change between two points. A slope of -1/3 means the line falls 1 unit vertically for every 3 units it moves horizontally to the right.

Point-Slope Form of a Line

The point-slope form, y - y₁ = m(x - x₁), is used to write the equation of a line when a point (x₁, y₁) and slope m are known. It helps in finding other points on the line by substituting values for x or y.

Plotting Points on a Coordinate Plane

Plotting points involves marking specific (x, y) coordinates on the Cartesian plane. To graph a line, you plot the given point and use the slope to find a second point by moving horizontally and vertically according to the slope.