Final Answer:
The parametric equations for the line passing through the points (x₁, y₁) = (2, 3) and (x₂, y₂) = (5, -1) are x = 2 + 3t and y = 3 - 4t.
Step-by-step explanation:
To determine the parametric equations for the line passing through the given points (x₁, y₁) = (2, 3) and (x₂, y₂) = (5, -1), first, find the slope of the line using the formula for slope, m = (y₂ - y₁) / (x₂ - x₁). Substituting the given coordinates yields m = (-1 - 3) / (5 - 2) = -4 / 3, representing the slope of the line.
The point-slope form of the equation of a line is y - y₁ = m(x - x₁). Plugging in the values of slope (m) and one of the points (x₁, y₁) = (2, 3) gives the equation y - 3 = (-4/3)(x - 2). This equation can be rearranged to y = 3 - (4/3)(x - 2), which simplifies to y = 3 - (4/3)x + 8/3.
Now, express x in terms of a parameter 't' to establish the parametric equations. Set x - 2 = 3t (isolating x - 2 to be equal to the parameter), which results in x = 2 + 3t. Substituting this expression for x into the equation for y yields y = 3 - (4/3)(2 + 3t) + 8/3. Simplifying further gives y = 3 - 8/3 - 4t + 8/3, leading to y = 3 - 8/3 - 4t + 8/3, and finally y = 3 - 4t.
Hence, the parametric equations for the line passing through the points (2, 3) and (5, -1) are x = 2 + 3t and y = 3 - 4t, where 't' represents any real number, defining all points on the line.