X/a+y/b=a+b , x/a2+y/b2=2

1.	X/a+y/b=a+b , x/a2+y/b2=2
Answer» $\large\underline{\bold{Given \:Question - }}$ 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 $\rm :\longmapsto\: \sf \: \dfrac{x}{ {a}^{2} } + \dfrac{\cancel{ {b}^{2}}\: \: ^{1} }{ \cancel{{b}^{2} } } = 2$ $\rm :\longmapsto\:\dfrac{x}{ {a}^{2} } + 1 = 2$ $\rm :\longmapsto\:\dfrac{x}{ {a}^{2} } = 1$ $\bf\implies \:x \: = \: {a}^{2}$ $\rm \: Hence, \: the \: solution \: is \: x = {a}^{2} \: \: and \: \: y = {b}^{2}$ ─━─━─━─━─━─━─━─━─━─━─━─━─ 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

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