

Answers
99x+101y=499-----(1)
101x+99y=501------(2)
Since the LCM of 99 and 101 is 9999, we need multiply (1) by
101 and (2) by 99,
9999x+10201y=50399------(3)
9999x+9801y=49599------(4)
Subtracting (4) from (3),
400y=800
➡y=2
Substituting y=2 in (1),
99x+101(2)=499
99x=499-202
99x=297
➡x=3
Hence,
↪x=3
↪y=2
Given Equations :
★ 99x + 101y = 499 ----------------- [1]
★ 101x + 99y = 501 ----------------- [2]
Equation [1] can be written as :
★ (100 - 1)x + (100 + 1)y = 499
Equation [2] can be written as :
★ (100 + 1)x + (100 - 1)y = 501
Multiplying Equation [1] with (100 + 1), We get :
(100 + 1)(100 - 1)x + (100 + 1)²y = 499(100 + 1)
Using Identity : (a + b)(a - b) = (a² - b²)
(100² - 1²)x + (101)²y = 499 × 101
(100² - 1)x + (101)²y = 499 × 101 --------- [3]
Multiplying Equation [2] with (100 - 1), We get :
(100 + 1)(100 - 1)x + (100 - 1)²y = 501(100 - 1)
(100² - 1²)x + (99)²y = 501 × 99
(100² - 1)x + (99)²y = 501 × 99 --------- [4]
Subtracting Equation [4] from Equation [3], We get :
(100² - 1)x + (101)²y - [(100² - 1)x + (99)²y] =
(499 × 101) - (501 × 99)
(100² - 1)x + (101)²y - (100² - 1)x - (99)²y =
(499 × 101) - (501 × 99)
(101)²y - (99)²y = 499(100 + 1) - 501(100 - 1)
y[101² - 99²] = (499 × 100) + 499 - (501 × 100) + 501
y[(100 + 1)² - (100 - 1)²] = 100(499 - 501) + 1000
Using Identity : (a + b)² - (a - b)² = 4ab
y[4(100)] = 100(-2) + 1000
y[400] = -200 + 1000
400y = 800
y = 2
Substituting y = 2 in Equation [1], We get :
99x + 101(2) = 499
99x + 202 = 499
99x = 499 - 202
99x = 297
x = 3
Answers :
★ x = 3
★ y = 2