Final Answer:
The equation of the straight line passing through the points (2,2), (4,5), and (6,8) is y = x + 1.
Step-by-step explanation:
To determine the equation of a straight line passing through the given points (2,2), (4,5), and (6,8), we use the point-slope form of the equation of a line, which is y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line, and m is the slope.
First, consider the slope between the points (2,2) and (4,5). The slope formula m = (y₂ - y₁) / (x₂ - x₁) can be applied: m = (5 - 2) / (4 - 2) = 3 / 2. Therefore, the slope between these two points is 3/2.
Next, use the point-slope form using the point (2,2) and the slope 3/2: y - 2 = 3/2(x - 2). This equation represents the line through the point (2,2) with the slope of 3/2.
To verify the equation, check if it holds for the third point (6,8). Substituting x = 6 into the equation y - 2 = 3/2(x - 2), we get y - 2 = 3/2(6 - 2) = 3/2 * 4 = 6. Thus, y - 2 = 6 implies y = 6 + 2 = 8, which matches the y-coordinate of the point (6,8).
Finally, simplifying the equation y - 2 = 3/2(x - 2) gives y = 3/2x - 3 + 2 = 3/2x - 1. Hence, the equation of the straight line passing through the points (2,2), (4,5), and (6,8) is y = 3/2x - 1, which can be rewritten as y = x + 1 by multiplying both sides of the equation by 2/2 to get rid of the fraction.