Question

solve the equation ax upon b-bx upon a =a^2-b^2​

Answer 1

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

               (a*x/b)-(b*x/a)-(((a+b)^2)/a*b)=0

Step-by-step explanation:

Step by step solution :

Step  1  :

           (a + b)2

Simplify   ————————

              a    

Equation at the end of step  1  :

     x     x    (a+b)2

 ((a•—)-(b•—))-(——————•b)  = 0

     b     a      a  

Step  2  :

Equation at the end of step  2  :

     x     x   b•(a+b)2

 ((a•—)-(b•—))-————————  = 0

     b     a      a    

Step  3  :

           x

Simplify   —

           a

Equation at the end of step  3  :

     x     x   b•(a+b)2

 ((a•—)-(b•—))-————————  = 0

     b     a      a    

Step  4  :

           x

Simplify   —

           b

Equation at the end of step  4  :

       x     xb     b • (a + b)2

 ((a • —) -  ——) -  ————————————  = 0

       b     a           a      

Step  5  :

Calculating the Least Common Multiple :

5.1    Find the Least Common Multiple

     The left denominator is :       b

     The right denominator is :       a

                 Number of times each Algebraic Factor

           appears in the factorization of:    Algebraic    

   Factor      Left

Denominator   Right

Denominator   L.C.M = Max

{Left,Right}

a  0 1 1

b  1 0 1

     Least Common Multiple:

     ab

Calculating Multipliers :

5.2    Calculate multipliers for the two fractions

   Denote the Least Common Multiple by  L.C.M

   Denote the Left Multiplier by  Left_M

   Denote the Right Multiplier by  Right_M

   Denote the Left Deniminator by  L_Deno

   Denote the Right Multiplier by  R_Deno

  Left_M = L.C.M / L_Deno = a

  Right_M = L.C.M / R_Deno = b

Making Equivalent Fractions :

5.3      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)2   and  (y2+y)/(y+1)3  are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

  L. Mult. • L. Num.      ax • a

  ——————————————————  =   ——————

        L.C.M               ab  

  R. Mult. • R. Num.      xb • b

  ——————————————————  =   ——————

        L.C.M               ab  

Adding fractions that have a common denominator :

5.4       Adding up the two equivalent fractions

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

ax • a - (xb • b)     a2x - xb2

—————————————————  =  —————————

       ab                ab    

Equation at the end of step  5  :

 (a2x - xb2)    b • (a + b)2

 ——————————— -  ————————————  = 0

     ab              a      

Step  6  :

Step  7  :

Pulling out like terms :

7.1     Pull out like factors :

  a2x - xb2  =   x • (a2 - b2)

Trying to factor as a Difference of Squares :

7.2      Factoring:  a2 - b2

Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =

        A2 - AB + BA - B2 =

        A2 - AB + AB - B2 =

        A2 - B2

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check :  a2  is the square of  a1

Check :  b2  is the square of  b1

Answer 2

Required Answer:-

Given:

• $\rm \dfrac{ax}{b} - \dfrac{bx}{a} = {a}^{2} - {b}^{2}$

To Find:

• The value of x.

Solution:

We have,

$\rm \implies \dfrac{ax}{b} - \dfrac{bx}{a} = {a}^{2} - {b}^{2}$

LCM of a and b is ab. Therefore,

$\rm \implies \dfrac{ {a}^{2} x - {b}^{2}x }{ab} = {a}^{2} - {b}^{2}$

Taking x as common in numerator part, we get,

$\rm \implies \dfrac{ x({a}^{2} - {b}^{2})}{ab} = {a}^{2} - {b}^{2}$

Now, cancel out a² - b² from both sides,

$\rm \implies \dfrac{ x}{ab} =1$

$\rm \implies x = ab$

★ Hence, the value of x is ab (x = ab)

Answer:

• x = ab