The given task asks to determine the equation of a line passing through two points $A(-1, 3)$ and $B(3, 1)$, and to decide whether the line is increasing or decreasing.

Solution:

1. Slope Calculation:
   The slope $m$ of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the points $A(-1, 3)$ and $B(3, 1)$: $m = \frac{1 - 3}{3 - (-1)} = \frac{-2}{4} = -\frac{1}{2}$

2. Equation of the Line:
   Using the point-slope form of a line: $y - y_1 = m(x - x_1)$ Substitute $m = -\frac{1}{2}$ and one of the points, say $A(-1, 3)$: $y - 3 = -\frac{1}{2}(x - (-1))$ Simplify: $y - 3 = -\frac{1}{2}(x + 1)$ $y - 3 = -\frac{1}{2}x - \frac{1}{2}$ $y = -\frac{1}{2}x + \frac{5}{2}$

   The equation of the line is: $y = -\frac{1}{2}x + \frac{5}{2}$

3. Increasing or Decreasing:
   Since the slope $m = -\frac{1}{2}$ is negative, the line is decreasing.

Final Answer:

• The equation of the line is: $y = -\frac{1}{2}x + \frac{5}{2}$
• The line is decreasing.

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Related Questions:

1. How can we graph the line using this equation?
2. How do we derive the equation if given only the slope and one point?
3. How can we confirm the slope visually on a graph?
4. What does the $y$-intercept $\frac{5}{2}$ represent geometrically?
5. What would the equation look like if the slope were positive?

Tip:

Always check the sign of the slope ($m$) to quickly determine whether a line is increasing (positive $m$) or decreasing (negative $m$).