Question

(ax + by)(cx - dy)

In the expression above, a, b, c, and d are non-zero constants and ad = bc. If ac = 18 and bd = 50, what is the value of the coefficient of the xy term when the expression is multiplied out and the like terms are collected?
A) 180
B) 90
C) 45
D) 225

Answer 1

The coefficient of the xy term after expanding the algebraic expression and applying the conditions ad = bc, ac = 18, and bd = 50 is -90, answer choice B.

Step-by-step explanation:

The student has presented an algebraic expression (ax + by)(cx - dy) and provided conditions ad = bc, ac = 18, and bd = 50. To find the value of the coefficient of the xy term when this expression is expanded, we multiply the two binomials. The xy term specifically comes from the product of ax and -dy, and the product of by and cx. Hence, the coefficient of the xy term will be -ady + bcy. Substituting the given conditions, we see -ady + bcy becomes -a(bc) + b(ac) due to the condition ad = bc. Then, using ac = 18 and bc = ad = bd/50, we can further simplify the coefficient to -18(d) + 18(b). Since we know bd = 50, we can let d = 50/b and simplify the equation to -18(50/b) + 18(b) = -900/b + 18b. Now, we are looking for the value of b such that -900/b + 18b = 0, which occurs when b² = 900/18 or b = 5. Therefore, the coefficient of the xy term becomes -900/5 + 18*5 = -180 + 90, which results in a coefficient of -90.