Some values of x and their corresponding values of y are shown in the table, where a is a constant. If there is a linear relationship between x and y, which of the following equations represents the relationship?

Explanation

Choice A is correct. The slope-intercept form of a linear equation in the xy-plane is y = mx + b, where m is the slope of the graph of the equation and b is the y-coordinate of the y-intercept of the graph. Any two ordered pairs (x1, y1) and (x2, y2) that satisfy a linear equation can be used to compute the slope of the graph of the equation using the formula m = y2 − y1 / x2 − x1. Substituting the two pairs (a, 0) and (3a, −a) from the table into the formula gives m = −a − 0/ 3a − a, which can be rewritten as −a/ 2a, or − 1/2. Substituting this value for m in the slope-intercept form of the equation produces y = -1/2 x + b. Substituting values from the ordered pair (a, 0) in the table into this equation produces 0 = -1/2(a) + b, which simplifies to b = a/2. Substituting this value for b in the slope-intercept form of the equation produces y = −1/2 x + a/2. Rewriting this equation in standard form by adding 1/2x to both sides and then multiplying both sides by 2 gives the equation x + 2y = a.