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need help with understanding this equation (x+1)y=y+3

MathematicsMathematics
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Lucy Davitson Posted

need help with understanding this equation (x+1)y=y+3

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            3 Simplify   —            4

Equation at the end of step  1  :

  (y - 1)          3  ——————— -  (0 -  —)  = 0   (x + 3)          4

Step  2  :

            y - 1 Simplify   —————            x + 3

Equation at the end of step  2  :

  (y - 1)    -3  ——————— -  ——  = 0    x + 3     4 

Step  3  :

Calculating the Least Common Multiple :

 3.1    Find the Least Common Multiple

      The left denominator is :       x+3 

      The right denominator is :       4 

        Number of times each prime factor
        appears in the factorization of:

 Prime 
 Factor 
 Left 
 Denominator 
 Right 
 Denominator 
 L.C.M = Max 
 {Left,Right} 
2 0 2 2
 Product of all 
 Prime Factors 
1 4 4
                  Number of times each Algebraic Factor
            appears in the factorization of:

    Algebraic    
    Factor    
 Left 
 Denominator 
 Right 
 Denominator 
 L.C.M = Max 
 {Left,Right} 
 x+3  1 0 1

      Least Common Multiple:
      4 • (x+3) 

Calculating Multipliers :

 3.2    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 = 4

   Right_M = L.C.M / R_Deno = x+3
 

Making Equivalent Fractions :

 3.3      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.      (y-1) • 4   ——————————————————  =   —————————         L.C.M             4 • (x+3)   R. Mult. • R. Num.      -3 • (x+3)   ——————————————————  =   ——————————         L.C.M             4 • (x+3) 

Adding fractions that have a common denominator :

 3.4       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:

 (y-1) • 4 - (-3 • (x+3))     4y + 3x + 5 ————————————————————————  =  ———————————        4 • (x+3)             4 • (x + 3)

Equation at the end of step  3  :

  4y + 3x + 5  ———————————  = 0   4 • (x + 3)

Step  4  :

When a fraction equals zero :

 4.1    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:

  4y+3x+5  ——————— • 4•(x+3) = 0 • 4•(x+3)  4•(x+3)

Now, on the left hand side, the  4 • x+3  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   4y+3x+5  = 0

Equation of a Straight Line

 4.2     Solve   4y+3x+5  = 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  4y+3x+5  = 0 and calculate its properties

Graph of a Straight Line :

   
  

Calculate the Y-Intercept :

Notice that when x = 0 the value of y is -5/4 so this line “cuts” the y axis at y=-1.25000

  y-intercept = -5/4  = -1.25000 

Calculate the X-Intercept :

When y = 0 the value of x is -5/3 Our line therefore “cuts” the x axis at x=-1.66667

  x-intercept = -5/3  = -1.66667 

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 -1.250 and for x=2.000, the value of y is -2.750. So, for a change of 2.000 in x (The change in x is sometimes referred to as “RUN”) we get a change of -2.750 – (-1.250) = -1.500 in y. (The change in y is sometimes referred to as “RISE” and the Slope is m = RISE / RUN)

    Slope     = -1.500/2.000 = -0.750 

Geometric figure: Straight Line

  1.   Slope = -1.500/2.000 = -0.750
  2.   x-intercept = -5/3 = -1.66667
  3.   y-intercept = -5/4 = -1.25000
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