Question

Solve X/a + y/b = a + b
And x/a^2 + y/b^2 = 2

Answer 1

GIVEN

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

SOLUTION

We will be solving the equations using Elimination method.

$= > \frac{x}{a} + \frac{y}{b} = a + b \\ \\ taking \: lcm \\ \\ = > \frac{bx + ay}{ab} = a + b \: \\ \\ = > bx + ay = (a + b)ab \\ \\ = > bx + ay = {a}^{2} b + a {b}^{2} \: - - - - (1)$

Now,

$= > \frac{x}{ {a}^{2} } + \frac{y}{ {b}^{2} } = 2 \\ \\taking \: lcm \: \: \\ \\ = > \frac{{b}^{2}x + {a}^{2} y}{ {a}^{2} {b}^{2} } = 2 \\ \\ = > b {}^{2}x + a {}^{2}y = 2 {a}^{2} {b}^{2} \: - - - - (2)$

Now we will multiply the equation by a suitable constant to make the coefficients of any one of the variables equal so that we can eliminate it. Here, We will multiply bx in eq(1) by b so that it will give b²x like eq(2).

$b \times (bx + ay = {a}^{2}b + a {b}^{2} ) \\ 1 \times (b {}^{2} x + {a}^{2} y = 2a {}^{2} b {}^{2} ) \\ \\ \implies \\ {b}^{2} x + aby = {a}^{2} b {}^{2} + ab {}^{3} \\ {b}^{2} x + {a}^{2} y = 2 {a}^{2} {b}^{2} \\ - \: \: \: \: \: \: - \: \: \: \: \: \: = - \: \: \: \: \: \: (subtracting) \\ we \: get \\ \\ \cancel{ {{b }^{2} x}} \: + aby \: = {a}^{2} {b}^{2} + ab {}^{3} \\ \cancel{ - {b}^{2}x } - {a}^{2} y = - 2 {a}^{2} b {}^{2} \\ \\ \small = > ( aby - {a}^{2} y )= ({a}^{2} b {}^{2} + ab {}^{3} - 2 {a}^{2} b {}^{2} ) \\ \\ = > (ab - {a}^{2} )y =( ab {}^{3} - {a}^{2} b {}^{2} ) \\ \\ we \: will \: take \: common \: \\ \\ = > a(b - a)y = a {b}^{2} ( {b} - a) \\ \\ = > y \: = \frac{ab {}^{2}( \cancel{b - a} )}{a( \cancel{b - a})} \\ \\ = > y = \frac{a {b}^{2} }{a} \\ \\ = > y = { \frac{ \cancel {a} \: b {}^{2} }{ \cancel{a}} } \\ \\ \implies \: y \: = {b}^{2}$

Putting value of y in eq(2)

$\implies {b}^{2}x + {a}^{2} {b}^{2} = 2 {a}^{2} {b}^{2} \\ \\ \implies {b}^{2} x \: = 2 {a}^{2} b {}^{2} - {a}^{2} b {}^{2} \\ \\ \implies \: {b}^{2} x = {a}^{2} b {}^{2} \\ \\ \implies \: x = \frac{ {a}^{2}b {}^{2} }{ {b}^{2} } \\ \\ \implies \: x = \frac{ {a}^{2} \: \cancel{ {b}^{2} } }{ \cancel{b {}^{2} }} \\ \\ \implies \: x \: = {a}^{2}$

x = a²

y = b²

Hence, the value of x is a² and y is b².

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