solve for x and y x+y=17;x²+y²=169
Answers
X+y=17
X2+Y2=169
(17)2=169+2XY
289-169=2XY
120=2XY
XY=60
then x+y=17
XY=60then X=12
Y=5
Answer:
x = 5 & y = 12 is a solution,
&
x = 12 & y= 5 is another solution.
Step-by-step explanation:
Given :
To find the value of x & y if,
x + y = 17 ...(i)
&
x² + y² = 169 ...(ii)
Solution :
We know that,
(a + b)² = a² + 2ab + b² ..(iii)
&
(a - b)² = a² - 2ab + b² ..(iv)
Hence, by using these identities,.
⇒ x + y = 17
⇒ (x + y)² = (17)²
⇒ x² + 2xy + y² = 289 (by (iii) )
⇒ (x² + y²) + 2xy = 289
⇒ 169 + 2xy = 289
⇒ 2xy = 289 - 169
⇒ 2xy = 120 ...(v)
__
(x - y)² = x² - 2xy + y² ( by (iv) )
⇒ (x - y)² = (x² + y²) - 2xy
⇒ (x - y)² = 169 - 120
⇒ (x - y)² = 49
⇒ x - y = ±7 (let it be +7)
⇒ x - y = 7 ...(vi)
By adding (i) & (vi),
We get,
⇒ (x + y) + (x - y) = 17 + 7
⇒ 2x = 24
⇒ x = 12
By substituting the value of x in (i),
We get,
⇒ x + y = 17
⇒ 12 + y = 17
⇒ y = 17 - 12
⇒ y = 5
∴ x = 12 & y = 5
and if x - y = -7, ..(vii)
Then,
By adding (vii) & (i),
We get,
⇒ (x + y) + (x - y) = 17 + (-7)
⇒ 2x = 10
⇒ x = 5,
By substituting the value of x in (i),
We get,.
⇒ x + y = 17
⇒ 4 + y = 17
⇒ y = 17 - 5
⇒ y = 12,.
x = 5 & y = 12 is a solution,
&
x = 12 & y= 5 is another solution,.