Question

$99x + 101y = 499$
$101x + 99y = 501$

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Answer 1

$\huge \bold{hey \: there}$

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

Answer 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 :

$:\implies$  (100 + 1)(100 - 1)x + (100 + 1)²y = 499(100 + 1)

Using Identity : (a + b)(a - b) = (a² - b²)

$:\implies$  (100² - 1²)x + (101)²y = 499 × 101

$:\implies$  (100² - 1)x + (101)²y = 499 × 101 --------- [3]

Multiplying Equation [2] with (100 - 1), We get :

$:\implies$  (100 + 1)(100 - 1)x + (100 - 1)²y = 501(100 - 1)

$:\implies$  (100² - 1²)x + (99)²y = 501 × 99

$:\implies$  (100² - 1)x + (99)²y = 501 × 99 --------- [4]

Subtracting Equation [4] from Equation [3], We get :

$:\implies$  (100² - 1)x + (101)²y - [(100² - 1)x + (99)²y] =

                                                                      (499 × 101) - (501 × 99)

$:\implies$  (100² - 1)x + (101)²y - (100² - 1)x - (99)²y =

                                                                      (499 × 101) - (501 × 99)

$:\implies$  (101)²y - (99)²y = 499(100 + 1) - 501(100 - 1)

$:\implies$  y[101² - 99²] = (499 × 100) + 499 - (501 × 100) + 501

$:\implies$  y[(100 + 1)² - (100 - 1)²] = 100(499 - 501) + 1000

Using Identity : (a + b)² - (a - b)² = 4ab

$:\implies$  y[4(100)] = 100(-2) + 1000

$:\implies$  y[400] = -200 + 1000

$:\implies$  400y = 800

$\mathsf{:\implies y = \dfrac{800}{400}}$

$:\implies$  y = 2

Substituting y = 2 in Equation [1], We get :

$:\implies$  99x + 101(2) = 499

$:\implies$  99x + 202 = 499

$:\implies$  99x = 499 - 202

$:\implies$  99x = 297

$\mathsf{:\implies x = \dfrac{297}{99}}$

$:\implies$  x = 3

Answers :

★  x = 3

★  y = 2