Final answer:
To determine which point lies on the graph of the linear equation passing through (4,1) and (7, -2), we calculate the slope, derive the equation of the line, and then test which point satisfies the equation. The correct point is (3,2).
Step-by-step explanation:
The task is to find which point lies on the graph of a linear equation that contains the points (4,1) and (7, -2). First, we need to calculate the slope of the line passing through these points. The slope m, given two points (x1, y1) and (x2, y2), is calculated by (y2 - y1) / (x2 - x1). In this case, using the given points (4,1) and (7, -2), the slope is (-2 - 1) / (7 - 4) = -3 / 3 = -1.
With the slope, we can now generate the equation of the line in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. However, to find b, we can substitute one of the points into the equation. Let's use (4,1): 1 = (-1)(4) + b. This gives us b = 1 + 4 = 5.
The equation of our line is y = -1x + 5. Now, to determine which of the provided points lies on this line, we substitute x from each point into the equation and see if the corresponding y value matches. Point (3,2) gives us y = -1(3) + 5, which equals 2, so point (3,2) lies on the graph of our linear equation.