Theory question 107
Linear and Simultaneous Equations(a) The sum of three numbers is 81. The second number is twice the first. Given that the third number is 6 more than the second, find the numbers.
(b) Given the points P(3, 5) and Q(−5, 7) on the cartesian plane such that R(x, y) is the midpoint of PQ, find the equation of the line that passes through R and perpendicular to PQ.
Show the model answer
Model answer
(a) Let the numbers be x, 2x, 2x + 6: 5x + 6 = 81 ⇒ x = 15. The numbers are 15, 30 and 36
(b) R = ((3 − 5)/2, (5 + 7)/2) = (−1, 6)
Gradient of PQ = (7 − 5)/(−5 − 3) = −¼, so the perpendicular gradient = 4
y − 6 = 4(x + 1) ⇒ y = 4x + 10
Explanation
Perpendicular gradients multiply to −1.