Question

x/a+y/b=a+b , x/a2+y/b2=2​

Answer 1

$\large\underline{\bold{Given \:Question - }}$

Solve the following pair of linear equations :-

$\sf \: \dfrac{x}{a} + \dfrac{y}{b} = a + b$

and

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

$\large\underline{\bold{Solution-}}$

There are three methods to solve such pair of linear equations

• 1. Method of Eliminations

• 2. Method of Substitution

• 3. Method of Cross multiplication

Prefered here Elimination method to solve the pair of linear equations.

The Elimination Method

Step 1: Multiply each equation by a suitable number so that the two equations have the same leading coefficient.

Step 2: Subtract the second equation from the first.

Step 3: Solve this new equation for one variable.

Step 4: Substitute the value of this variable into either Equation 1 or Equation 2 to get the value of second variable.

Let's solve the problem now !!!

The given pair of linear equations are

$\sf \: \dfrac{x}{a} + \dfrac{y}{b} = a + b - - - (1)$

and

$\sf \: \dfrac{x}{ {a}^{2} } + \dfrac{y}{ {b}^{2} } = 2 - - - (2)$

The equation (1) can be simplified as

$\rm :\longmapsto\:\dfrac{bx + ay}{ab} = a + b$

$\rm :\implies\:bx + ay = b {a}^{2} + {ab}^{2} - - - (3)$

The equation (2) can be simplified as

$\rm :\longmapsto\:\dfrac{ {xb}^{2} + {ya}^{2} }{ {a}^{2} {b}^{2} } = 2$

$\rm :\implies\: {xb}^{2} + {ya}^{2} = 2 {a}^{2} {b}^{2} - - - (4)$

Now, Multiply equation (3) by b, we get

$\rm :\longmapsto\: {xb}^{2} + aby = {b}^{2} {a}^{2} + {ab}^{3} - - - (5)$

Now, Subtracting equation (5) from equation (4) we get

$\rm :\longmapsto\: {ya}^{2} - aby = {a}^{2} {b}^{2} - {ab}^{3}$

$\rm :\longmapsto\:y \: \cancel{a} \: \cancel{(a - b)} = {\cancel{a} \: b}^{2} \: \cancel{ (a - b)}$

$\bf\implies \:y \: = \: {b}^{2} - - (6)$

Substituting the value of y from equation (6) in (2), we get

$\rm :\longmapsto\: \sf \: \dfrac{x}{ {a}^{2} } + \dfrac{\cancel{ {b}^{2}}\: \: ^{1} }{ \cancel{{b}^{2} } } = 2$

$\rm :\longmapsto\:\dfrac{x}{ {a}^{2} } + 1 = 2$

$\rm :\longmapsto\:\dfrac{x}{ {a}^{2} } = 1$

$\bf\implies \:x \: = \: {a}^{2}$

$\rm \: Hence, \: the \: solution \: is \: x = {a}^{2} \: \: and \: \: y = {b}^{2}$

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Additional Information :-

Method of Substitution

To solve systems using substitution, follow this procedure:

• Select one equation and solve it for one of its variables.

• In the other equation, substitute for the variable just solved.

• Solve the new equation.

• Substitute the value found into any equation involving both variables and solve for the other variable