Solve each of the following systems of equations by the method of cross-multiplication :
$x\left(a-b+\frac{a b}{a-b}\right)=y\left(a+b-\frac{a b}{a+b}\right)$
$x+y=2 a^{2}$
GIVEN:
$x\left((a-b)+\frac{a b}{a-b}\right)=y\left((a+b)-\frac{a b}{a+b}\right)$
$x+y=2 a^{2}$
To find: The solution of the systems of equation by the method of cross-multiplication:
Here we have the pair of simultaneous equation
$x\left((a-b)+\frac{a b}{a-b}\right)-y\left((a+b)-\frac{a b}{a+b}\right)=0$
$x+y-2 a^{2}=0$
By cross multiplication method we get
$\frac{x}{\left(\left(-2 a^{2}\right) \times-\left((a+b)-\frac{a b}{a+b}\right)\right)-0}=\frac{-y}{\left(-2 a^{2}\right) \times\left((a-b)+\frac{a b}{a-b}\right)-0}$
$=\frac{1}{\left((a-b)+\frac{a b}{a-b}\right)-\left(-\left((a+b)-\frac{a b}{a+b}\right)\right)}$
$\frac{x}{\left(\left(-2 a^{2}\right) \times-\left(\frac{(a+b)^{2}-a b}{a+b}\right)\right)}=\frac{-y}{\left(-2 a^{2}\right) \times\left(\frac{(a-b)^{2}+a b}{a-b}\right)}$
$=\frac{1}{\left(\frac{(a-b)^{2}+a b}{a-b}\right)-\left(-\left(\frac{(a+b)^{2}-a b}{a+b}\right)\right)}$
$\frac{x}{\left(\left(-2 a^{2}\right) \times-\left(\frac{\left(a^{2}+b^{2}+2 a b\right)-a b}{a+b}\right)\right)}=\frac{-y}{\left(-2 a^{2}\right) \times\left(\frac{\left(a^{2}+b^{2}-2 a b\right)+a b}{a-b}\right)}$
$=\frac{1}{\left(\frac{\left(a^{2}+b^{2}-2 a b\right)+a b}{a-b}\right)-\left(-\left(\frac{\left(a^{2}+b^{2}+2 a b\right)-a b}{a+b}\right)\right)}$
$\frac{x}{\left(\left(-2 a^{2}\right) \times-\left(\frac{\left(a^{2}+b^{2}+a b\right)}{a+b}\right)\right)}=\frac{-y}{\left(-2 a^{2}\right) \times\left(\frac{\left(a^{2}+b^{2}-a b\right)}{a-b}\right)}$
$=\frac{1}{\left(\frac{\left(a^{2}+b^{2}-a b\right)}{a-b}\right)-\left(-\left(\frac{\left(a^{2}+b^{2}+a b\right)}{a+b}\right)\right)}$
$\frac{\frac{x}{\left(2 a^{4}+2 a^{2} b^{2}+2 a^{3} b\right)}}{a+b}=\frac{y}{\frac{\left(2 a^{4}+2 a^{2} b^{2}-2 a^{3} b\right)}{a-b}}$
$=\frac{1}{\left(\frac{\left(a^{2}+b^{2}-a b\right)}{a-b}\right)-\left(-\left(\frac{\left(a^{2}+b^{2}+a b\right)}{a+b}\right)\right)}$
$\frac{x}{\frac{\left(2 a^{4}+2 a^{2} b^{2}+2 a^{3} b\right)}{a+b}}=\frac{y}{\frac{\left(2 a^{4}+2 a^{2} b^{2}-2 a^{3} b\right)}{a-b}}$
$=\frac{1}{\left(\frac{\left(a^{2}+b^{2}-a b\right)(a+b)+\left(a^{2}+b^{2}+a b\right)(a-b)}{(a-b)(a+b)}\right)}$
$\frac{x}{\frac{\left(2 a^{4}+2 a^{2} b^{2}+2 a^{3} b\right)}{a+b}}=\frac{\frac{1}{\left(2 a^{4}+2 a^{2} b^{2}-2 a^{3} b\right)}}{a-b}=\frac{1}{\left(\frac{\left(a^{3}+b^{3}+a^{3}-b^{3}\right)}{(a-b)(a+b)}\right)}$
$\frac{x}{\frac{\left(2 a^{4}+2 a^{2} b^{2}+2 a^{3} b\right)}{a+b}}=\frac{y}{\frac{\left(2 a^{4}+2 a^{2} b^{2}-2 a^{3} b\right)}{a-b}}=\frac{1}{\left(\frac{2 a^{3}}{(a-b)(a+b)}\right)}$
Consider the following for x
$\frac{x}{\frac{\left(2 a^{4}+2 a^{2} b^{2}+2 a^{3} b\right)}{a+b}}=\frac{1}{\left(\frac{2 a^{3}}{(a-b)(a+b)}\right)}$
$\frac{x}{\frac{\left(a^{2}+b^{2}+a b\right)}{a+b}}=\frac{1}{\left(\frac{a}{(a-b)(a+b)}\right)}$
$x\left(\frac{a}{(a-b)(a+b)}\right)=\frac{\left(a^{2}+b^{2}+a b\right)}{a+b}$
$x=\frac{\left(a^{2}+b^{2}+a b\right)(a-b)}{a}$
$x=\frac{\left(a^{3}+a b^{2}+a^{2} b-b^{3}-a b^{2}-a^{2} b\right)}{a}$
$x=\frac{\left(a^{3}-b^{3}\right)}{a}$
And
$\frac{y}{\frac{\left(2 a^{4}+2 a^{2} b^{2}-2 a^{3} b\right)}{a-b}}=\frac{1}{\left(\frac{2 a^{3}}{(a-b)(a+b)}\right)}$
$\frac{y}{\frac{\left(a^{2}+b^{2}-a b\right)}{a-b}}=\frac{1}{\left(\frac{a}{(a-b)(a+b)}\right)}$
$\frac{y}{\frac{\left(a^{2}+b^{2}-a b\right)}{a-b}}=\frac{1}{\left(\frac{a}{(a-b)(a+b)}\right)}$
$y\left(\frac{a}{(a-b)(a+b)}\right)=\frac{\left(a^{2}+b^{2}-a b\right)}{a-b}$
$y=\frac{\left(a^{2}+b^{2}-a b\right)(a+b)}{a}$
$y=\frac{\left(a^{3}+b^{3}\right)}{a}$
Hence we get the value of $x=\frac{a^{3}-b^{3}}{a}$ and $y=\frac{a^{3}+b^{3}}{a}$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,