Mathematics

Problem

Let

x_1, x_2, x_3, \ldots, x_n

be the solutions of the

n

equations

x_1 + b_1 = c_1

x_2 + b_2 = c_2

, \ldots,

x_n + b_n = c_n

, where each

b_i

and

c_i

is a real number. a. (i) Show that

x_i = c_i - b_i

for each

i

. b. (ii) Hence prove that

\displaystyle \sum_{i=1}^{n} x_i = \sum_{i=1}^{n} c_i - \sum_{i=1}^{n} b_i

. c. (iii) Use the result of part (ii) to find the sum of the solutions of the following five equations without solving them one by one: (a)

x + 3.2 = 7.9

(b)

y - 4.5 = -1.5

(c)

z + (-2) = 0.8

(d)

u - 6 = -10

(e)

v + 9 = 4

. d. (iv) Verify your answer in part (iii) by completely solving any ONE of the five equations and confirming that the sum remains unchanged.