Learning Combined Equation

Learning Combined equation deals with an equation in which a homogeneous function is equal to zero. Combined equation relates with the Homogeneous function in which through the equation, in the given way as like the sum of the indices occurring in each term is constant. For e.g. x³ + 2x² + y³ +5xy², here degree of each term is 3. by this way learning of combined equation is proceeded.

 

Learning Combined equation – Definition

 

A combined equation of the two lines all the way through the origin is a homogenous equation of second degree. – Let the two strokes overtake through the starting point be y = m 1 x and y = m 2 x. i.e. y – m 1 x = 0 and y – m 2 x = 0 their combined equation then this is clearly a homogeneous or combined equation… ²A normal, integral, algebraic equation of the variables x and y is said to be homogeneous equation of n^th degree in x and y, when the sum of the indices of x and y in every term is the same and is equal to n.

I am planning to write more post on Inverse Functions Calculus with example, How to Solve Multi Step Inequalities. Keep checking my blog.

 

Example for learning combined equation

 

Combined equations of two outlines are in all the way through the origin as a homogeneous equation of the second degree

Let,

y = S₁x … (1)

y = S₂x … (2)

Be the lines through the origin.

The equations (1) and (2) can be written as

y – S₁ x = 0 and y – S₂x = 0

Their combined equation is

(y – S₁X)(Y- S2X) = 0

Y² – (S1 + S2)XY + S1S2X² = 0  …(3)

This is clearly a homogenous equation of the second degree in x and y.

To express ax² + 2hxy + by² = 0 as bS² + 2hS + a =0

ax² + 2hxy + by² = 0

Divide equation (1) by x²,

a+ 2h`(y)/(x)` + b(`(y)/(x)`)² = 0

Substituting `(y)/(x)` = S, we get

a+ 2hS + bS² = 0

This can be written as bS² + 2hS + a = 0

This is a quadratic form in m. This has two roots (say) S₁ and S_2.

Therefore S₁ + S₂  = `(-2h)/(b)` ,  S₁S₂ = `(a)/(b)`