Question

Solve: x/a+y/b=a+b ; x/a2 + y/b2=2

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{\dfrac{x}{a}+\dfrac{y}{b}=a+b}$

$\mathsf{\dfrac{x}{a^2}+\dfrac{y}{b^2}=2}$

$\underline{\textbf{To find:}}$

$\textsf{Solution of the given equations}$

$\underline{\textbf{Solution:}}$

$\textsf{The given equation can be written as}$

$\mathsf{\left(\dfrac{1}{a}\right)x+\left(\dfrac{1}{b}\right)y-(a+b)=0}$

$\mathsf{\left(\dfrac{1}{a^2}\right)x+\left(\dfrac{1}{b^2}\right)y-2=0}$

$\textsf{By Cross multiplication rule,}$

$\mathsf{\dfrac{x}{\dfrac{-2}{b}+\dfrac{a+b}{b^2}}=\dfrac{y}{\dfrac{-(a+b)}{a^2}+\dfrac{2}{a}}=\dfrac{1}{\dfrac{1}{ab^2}-\dfrac{1}{a^2b}}}$

$\mathsf{\dfrac{x}{\dfrac{-2b+a+b}{b^2}}=\dfrac{y}{\dfrac{-a-b+2a}{a^2}}=\dfrac{1}{\dfrac{a-b}{a^2b^2}}}$

$\mathsf{\dfrac{x}{\dfrac{a-b}{b^2}}=\dfrac{y}{\dfrac{a-b}{a^2}}=\dfrac{1}{\dfrac{a-b}{a^2b^2}}}$

$\begin{array}{c|c}\mathsf{\dfrac{x}{\dfrac{a-b}{b^2}}=\dfrac{1}{\dfrac{a-b}{a^2b^2}}}&\mathsf{\dfrac{y}{\dfrac{a-b}{a^2}}=\dfrac{1}{\dfrac{a-b}{a^2b^2}}}\\&\\\mathsf{\dfrac{xb^2}{a-b}=\dfrac{a^2b^2}{a-b}}&\mathsf{\dfrac{ya^2}{a-b}=\dfrac{a^2b^2}{a-b}}\\&\\\mathsf{x=\dfrac{a^2b^2}{a-b}{\times}\dfrac{a-b}{b^2}}&\mathsf{y=\dfrac{a^2b^2}{a-b}{\times}\dfrac{a-b}{a2}}\\&\\\mathsf{x=a^2}&\mathsf{y=b^2}\end{array}$

$\therefore\textbf{Solution\;\boldmath x=a^2\;\;\&\;\;y=b^2 }$

$\underline{\textbf{Find more:}}$

Solve each of the following systems of equations by the method of cross-multiplication:

ax + by = a²

bx + ay = b²

https://brainly.in/question/15918815  

Solve each of the following systems of equations by the method of cross-multiplication:

a²x+b²y=c²b²x+a²y=a²

https://brainly.in/question/15918831

Answer 2

x = a^2, y = b^2

Step-by-step explanation:

Given that,

x/a + y/b = a + b; x/a^2 + y/b^2 = 2

Solving the equation x/a + y/b = a + b

⇒ bx + ay = ab(a + b)  ...(i)

Solving the equation x/a^2 + y/b^2 = 2,

⇒ b^2x + a^2y = 2a^2b^2   ...(ii)

Now,

b * (i) - (ii)

⇒ [b^2x + aby] - [(b^2x + a^2y)] = ab^2(a + b) - 2a^2b^2

⇒  b^2x + aby - b^2x - a^2y = a^2b^2 + ab^3 - 2a^2b^2

⇒  aby - a^2y = ab^3 - a^2b^2

⇒ y (b-a)a = ab^2(b- a)

⇒  y = ab^2(b - a)/a(b-a)

∵   y = b^2

While putting the value of y in equation (ii),

b^2x + a^2y = 2a^2b^2

⇒ b^2x = 2a^2b^2 - a^2b^2

⇒ x = (a^2b^2)/b^2

∵ x = a^2

Learn more: Cross-multiplication

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