Final answer:
The student needs to determine the equation of a linear function that passes through two points and has a given slope. The solution involves applying the slope formula to the given points and matching the result with the provided algebraic expressions.
Step-by-step explanation:
The student is asking for assistance in determining the equation of a line that passes through two given points with a specific slope. The question is a Mathematics question at the High School level. To find an equation of a line, we generally use the slope-intercept form (y = mx + b), with m representing the slope and b the y-intercept. Since we don't have the y-intercept, we can alternatively use the point-slope form or just apply the slope formula directly, which states that for two points (x1, y1) and (x2, y2) with a slope m, the relation is (y2 - y1) / (x2 - x1) = m.
To solve this problem, we plug in the points (4, w) and (r, 7) into the slope formula. The slope is described by the options provided, which are not actual slopes but algebraic expressions. The goal is to manipulate the algebraic expressions to match the slope that we find using the coordinates provided. An example of a concrete slope would be similar to the ones given in the provided reference examples, such as the fixed slope of 3 in Figure A1, where for every increase of 1 on the x-axis, there is a rise of 3 on the y-axis.
Using the slope formula (7 - w) / (r - 4) = m, and knowing the options provided, we can now match the slope with the correct algebraic expression. We can equate this to each algebraic expression and solve for r and w, dismissing the irrelevant options to find the correct answer.