Solve: ax + by = a - b and bx - ay = a + b

1.	Solve: ax + by = a - b and bx - ay = a + b
Answer» Basic Concept :- There are 4 methods to SOLVE this type of pair of linear equations. 1. Method of Substitution 2. Method of Eliminations 3. Method of Cross Multiplication 4. GRAPHICAL Method We prefer here Method of Eliminations. To solve systems using elimination, follow this procedure: 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 to reduce the equation to one variable. Step 3: Solve this new equation for one variable. Step 4: Substitute the value of this variable into either of Equation 1 or Equation 2 above and solve for other variable. Let's solve the PROBLEM now!! Given equations are $\rm :\longmapsto\:ax + by = a - b - - (1)$ $\rm :\longmapsto\:bx - ay = a + b - - (2)$ On multiply equation (1) by b, we get $\rm :\longmapsto\:abx + {b}^{2} y = ab - {b}^{2} - - (3)$ On multiplying equation (2) by a, we get $\rm :\longmapsto\:abx - {a}^{2} y = {a}^{2} + ab - - (4)$ Subtracting equation (4) from equation (3), we get $\rm :\longmapsto\: {b}^{2} y + {a}^{2} y = - {b}^{2} - {a}^{2}$ $\rm :\longmapsto\:y \: \cancel{( {a}^{2} + {b}^{2})} = - \: \cancel{( {b}^{2} + {a}^{2})}$ $\bf\implies \:y \: = \: - \: 1$ Now, Substitute the value of y in equation (1), we get $\rm :\longmapsto\:ax - b = a - b$ $\rm :\longmapsto\:ax = a$ $\bf\implies \:x \: = \: 1$ $\overbrace{ \underline { \boxed { \bf \therefore \: The \:solution\: is \: x \: = 1 \: \: and \: \: y \: = \: - 1)}}}$

Answer»

Basic Concept :-

There are 4 methods to SOLVE this type of pair of linear equations.