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Rearrange:

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

           (4*x^2)/(x*y)+(9*y^2)/(x*y)-(15)=0 

Step  1  :

Equation at the end of step  1  :

   (4•(x2)) 32y2
  (————————+————)-15  = 0 
      xy     xy 

Step  2  :

            32y2
 Simplify   ————
             xy 

Dividing exponential expressions :

 2.1    y² divided by y¹ = y^{(2 - 1)} = y¹ = y

Equation at the end of step  2  :

   (4 • (x2))    9y     
  (—————————— +  ——) -  15  = 0 
       xy        x      
 Step  3  :

Equation at the end of step  3  :

   22x2    9y     
  (———— +  ——) -  15  = 0 
    xy     x      

Step  4  :

            22x2
 Simplify   ————
             xy 

Dividing exponential expressions :

 4.1    x² divided by x¹ = x^{(2 - 1)} = x¹ = x

Equation at the end of step  4  :

   4x    9y     
  (—— +  ——) -  15  = 0 
   y     x      

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       y 

      The right denominator is :       x 

      Least Common Multiple:
      xy 

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 = x

   Right_M = L.C.M / R_Deno = y

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.      4x • x
   ——————————————————  =   ——————
         L.C.M               xy  

   R. Mult. • R. Num.      9y • y
   ——————————————————  =   ——————
         L.C.M               xy  

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:

 4x • x + 9y • y     4x2 + 9y2
 ———————————————  =  —————————
       xy               xy    

Equation at the end of step  5  :

  (4x2 + 9y2)    
  ——————————— -  15  = 0 
      xy         

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  xy  as the denominator :

          15     15 • xy
    15 =  ——  =  ———————
          1        xy   

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 6.2       Adding up the two equivalent fractions

 (4x2+9y2) - (15 • xy)     4x2 - 15xy + 9y2
 —————————————————————  =  ————————————————
          xy                      xy       

Trying to factor a multi variable polynomial :

 6.3    Factoring    4x² - 15xy + 9y² 

Try to factor this multi-variable trinomial using trial and error 

 Found a factorization  :  (x - 3y)•(4x - 3y)

Equation at the end of step  6  :

  (x - 3y) • (4x - 3y)
  ————————————————————  = 0 
           xy         

Step  7  :

When a fraction equals zero :

 7.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

  (x-3y)•(4x-3y)
  —————————————— • xy = 0 • xy
        xy      

Now, on the left hand side, the  xy  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   (x-3y)  •  (4x-3y)  = 0

Theory - Roots of a product :

 7.2    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

Equation of a Straight Line

 7.3     Solve   x-3y  = 0

Tiger recognizes that we have here an equation of a straight line. Such an equation is usually written y=mx+b ("y=mx+c" in the UK).

"y=mx+b" is the formula of a straight line drawn on Cartesian coordinate system in which "y" is the vertical axis and "x" the horizontal axis.

In this formula :

y tells us how far up the line goes
x tells us how far along
m is the Slope or Gradient i.e. how steep the line is
b is the Y-intercept i.e. where the line crosses the Y axis

The X and Y intercepts and the Slope are called the line properties. We shall now graph the line  x-3y  = 0 and calculate its properties

Graph of a Straight Line :

Calculate the Y-Intercept :

Notice that when x = 0 the value of y is 0/-3 so this line "cuts" the y axis at y=-0.00000

  y-intercept = 0/-3  = -0.00000 

Calculate the X-Intercept :

When y = 0 the value of x is 0/1 Our line therefore "cuts" the x axis at x= 0.00000

  x-intercept = 0/1  =  0.00000 

Calculate the Slope :

Slope is defined as the change in y divided by the change in x. We note that for x=0, the value of y is 0.000 and for x=2.000, the value of y is 0.667. So, for a change of 2.000 in x (The change in x is sometimes referred to as "RUN") we get a change of 0.667 - 0.000 = 0.667 in y. (The change in y is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

    Slope     =  0.667/2.000 =  0.333 

Equation of a Straight Line

 7.4     Solve   4x-3y  = 0

Tiger recognizes that we have here an equation of a straight line. Such an equation is usually written y=mx+b ("y=mx+c" in the UK).

"y=mx+b" is the formula of a straight line drawn on Cartesian coordinate system in which "y" is the vertical axis and "x" the horizontal axis.

In this formula :

y tells us how far up the line goes
x tells us how far along
m is the Slope or Gradient i.e. how steep the line is
b is the Y-intercept i.e. where the line crosses the Y axis

The X and Y intercepts and the Slope are called the line properties. We shall now graph the line  4x-3y  = 0 and calculate its properties

Graph of a Straight Line :

Calculate the Y-Intercept :

Notice that when x = 0 the value of y is 0/-3 so this line "cuts" the y axis at y=-0.00000

  y-intercept = 0/-3  = -0.00000 

Calculate the X-Intercept :

When y = 0 the value of x is 0/4 Our line therefore "cuts" the x axis at x= 0.00000

  x-intercept = 0/4  =  0.00000 

Calculate the Slope :

Slope is defined as the change in y divided by the change in x. We note that for x=0, the value of y is 0.000 and for x=2.000, the value of y is 2.667. So, for a change of 2.000 in x (The change in x is sometimes referred to as "RUN") we get a change of 2.667 - 0.000 = 2.667 in y. (The change in y is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

    Slope     =  2.667/2.000 =  1.333 

Geometric figure: Two Straight Lines

1.   Slope = 2.667/2.000 = 1.333
2.   x-intercept = 0/4 = 0.00000
3.   y-intercept = 0/-3 = -0.00000
4.   Slope = 0.667/2.000 = 0.333
5.   x-intercept = 0/1 = 0.00000
6.   y-intercept = 0/-3 = -0.00000