Solve the following equations for x and y. log100 |x+y| = 1/2,

Question

Solve the following equations for x and y. 

log₁₀₀ |x+y| = \(\frac{1}{2}\), log₁₀ y – log₁₀ | x | = log₁₀₀4

(a) \(\bigg(\frac{8}{3},\frac{16}{3}\bigg)\), (–8, –16) 

(b) \(\bigg(\frac{10}{3},\frac{20}{3}\bigg)\). (+ 10, 20)

(c) \(\bigg(-\frac{10}{3},-\frac{20}{3}\bigg)\) (70, 20)

(d) None of these

Answer 1

(b) \(\bigg(\frac{10}{3},\frac{20}{3}\bigg)\). (+ 10, 20)

 log₁₀₀ |x+y| = \(\frac{1}{2}\) ⇒ |x + y| = 100^{\(^{\frac{1}{2}}\)}

⇒ |x + y| = 10 as (–10 is inadmissible)        ...(i) 

log₁₀y – log₁₀| x | = log₁₀₀4

⇒ log10 \(\frac{y}{|x|}\) = log_10² = log₁₀ 2         \(\big[\)Using log_aⁿ (x^m) = \(\frac{m}{n}\) log_a x\(\big]\)

⇒ \(\frac{y}{|x|}\) = 2 ⇒ y = 2 | x |                            ...(ii)

Substituting the value of y from (ii) in (i), we get 

| x + 2| x || = 10 

If x > 0, then 3x = 10 ⇒ x = \(\frac{10}{3}\)

If x < 0, then x = 10.

∴ If x = \(\frac{10}{3}\), then y = \(\frac{20}{3}\) and if x = 10, y = 20.