Question

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Solve the given equations.

=> x/a - y/b = 0

=> ax + by = a² + b²

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Answer 1

Answer:

Step-by-step explanation:

This can be done in 2 ways :- CND [Componendo - Divdendo Method] or Reduction method.

However, reduction method is preferable.

$\sf{\dfrac{x}{a} - \dfrac{y}{b} = 0}$

Simplifying the expression by taking LCM,

$\sf{\dfrac{bx - ay}{ab} = 0}$

$\sf{bx - ay= 0}$

Let this be equation (1)

Now, moving on to next,

$\sf{ax + by = a^2 + b^2}$ (2)

Now, as we will apply reduction, multiplying 1st variable with (1) and vice versa with (2)

$\sf{a(bx - ay= 0)}$

$\sf{abx - a^2 y= 0)}$  (4)

$\sf{b(ax + by = a^2 + b^2)}$

$\sf{bax + b^2 y = b(a^2 + b^2)}$ (3)

Now, take equations (3) and (4) and subtract them.

$\sf{bax + b^2 y - abx + a^2 y = b(a^2 + b^2)}$

Now, bax and abx get cancelled, because they are same terms in positive and negative, followed by taking y common.

$\sf{ y(b^2 + a^2) = b(a^2 + b^2)}$

(a² + b²) get cancelled on both sides, we get

y = b ------------ Answer 1.

Now, placing the value of y in equation (1),

$\sf{bx - ab= 0}$

$\sf{b(x - a)= 0}$

$\sf{x - a= 0}$

We get x = a --------- Answer 2.

Hence, x=a and y=b.

Answer 2

Required answer:-

Question:

Solve the given equations.

=> x/a - y/b = 0

=> ax + by = a² + b²

Solution:

Given,

★ x/a - y/b = 0

★ ax + by = a² + b²

Concept:

★ Linear equations

Understanding the concept.....

An equation of the form ax+by+c = 0 is called a linear equation in which a, b and c are constants (real numbers) and x and y are variables each with degree 1.

Consider the two linear equations: 3x + 4y = 6 and 8x + 5y = 3. These two equations contain same two variables (x and y in this case) . Together such equations are called linear equations.

Methods of solving linear equations:

★ Method of elimination by substitution.

★ Method of elimination by equating coefficients.

★ Method of cross multiplication.

Method used in the given equations:

★ Method of elimination by substitution.

Step by step explaination:

→ x/a - y/b = 0

→ x/a = y/b

→ x = a/b y ..(1)

As,

ax + by = a² + b² ...(2)

Now putting value of x in equation (2)

That is,

→ a (a/b y ) + by = a² + b²

→ a²y + b²y = b(a² + b²)

→ y (a² + b²) = b (a² + b²)

→ y = b

Now putting value of y in equation (1)

→ x = a/b × b

→ x = a

Answer:

x = a

y = b