Question:
c) Let's write the value of x³ + y³ by calculation if x + y = 2 and 1/x + 1/y = 2.
x³ + y³ = 2
Given that x + y = 2 and 1/x + 1/y = 2. We are asked to find out the value of x³ + y³.
[tex]\implies[/tex] 1/x + 1/y = 2
Take LCM. LCM of x & y is xy.
[tex]\implies[/tex] (y + x)/xy = 2
[tex]\implies[/tex] (x + y)/xy = 2 -------(eq 1)
Value of (x + y) is given. So, substitute the value of (x + y) = 2 in (eq 1).
[tex]\implies[/tex] 2/xy = 2
[tex]\implies[/tex] 2/xy = 2/1
Cross-multiply them,
[tex]\implies[/tex] 2 = 2xy
Divide by 2 on both sides,
[tex]\implies[/tex] 2/2 = 2xy/2
[tex]\implies[/tex] xy = 1
Now,
[tex]\implies[/tex] (x + y)³ = x³ + y³ + 3x²y + 3xy²
Used identity: (a + b)³ = a³ + b³ + 3a²b + 3ab²
[tex]\implies[/tex] (x + y)³ = x³ + y³ + 3xy(x + y) ---(eq 2)
Substitute the values in (eq 2)
[tex]\implies[/tex] (2)³ = x³ + y³ + 3(1)(2)
[tex]\implies[/tex] 2 × 2 × 2 = x³ + y³ + 6
[tex]\implies[/tex] 8 = x³ + y³ + 6
Subtract 6 on both sides,
[tex]\implies[/tex] 8 - 6 = x³ + y³ + 6 - 6
[tex]\implies[/tex] 2 = x³ + y³
[tex]\implies[/tex] x³ + y³ = 2
Hence, the value of x³ + y³ is 2.
(Algebraic Identities help in simplification)
For two variables:
For three variables:
Whereas algebraic identity for (x + a) (x + b) = x² + (a + b) x + ab.
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Question:
c) Let's write the value of x³ + y³ by calculation if x + y = 2 and 1/x + 1/y = 2.
Answer:
x³ + y³ = 2
Step-by-step explanation:
Given that x + y = 2 and 1/x + 1/y = 2. We are asked to find out the value of x³ + y³.
[tex]\implies[/tex] 1/x + 1/y = 2
Take LCM. LCM of x & y is xy.
[tex]\implies[/tex] (y + x)/xy = 2
[tex]\implies[/tex] (x + y)/xy = 2 -------(eq 1)
Value of (x + y) is given. So, substitute the value of (x + y) = 2 in (eq 1).
[tex]\implies[/tex] 2/xy = 2
[tex]\implies[/tex] 2/xy = 2/1
Cross-multiply them,
[tex]\implies[/tex] 2 = 2xy
Divide by 2 on both sides,
[tex]\implies[/tex] 2/2 = 2xy/2
[tex]\implies[/tex] xy = 1
Now,
[tex]\implies[/tex] (x + y)³ = x³ + y³ + 3x²y + 3xy²
Used identity: (a + b)³ = a³ + b³ + 3a²b + 3ab²
[tex]\implies[/tex] (x + y)³ = x³ + y³ + 3xy(x + y) ---(eq 2)
Substitute the values in (eq 2)
[tex]\implies[/tex] (2)³ = x³ + y³ + 3(1)(2)
[tex]\implies[/tex] 2 × 2 × 2 = x³ + y³ + 6
[tex]\implies[/tex] 8 = x³ + y³ + 6
Subtract 6 on both sides,
[tex]\implies[/tex] 8 - 6 = x³ + y³ + 6 - 6
[tex]\implies[/tex] 2 = x³ + y³
[tex]\implies[/tex] x³ + y³ = 2
Hence, the value of x³ + y³ is 2.
Algebraic Identities:
(Algebraic Identities help in simplification)
For two variables:
For three variables:
Whereas algebraic identity for (x + a) (x + b) = x² + (a + b) x + ab.