No solutions found

Rearrange:

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

                     x^2-(y^2)=0 

Step  1  :

Trying to factor as a Difference of Squares :

 1.1      Factoring:  x²-y² 

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 :  x²  is the square of  x¹ 

Check :  y²  is the square of  y¹ 

Factorization is :       (x + y)  •  (x - y) 

Equation at the end of step  1  :

  (x + y) • (x - y)  = 0 

Step  2  :

Theory - Roots of a product :

 2.1    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

 2.2     Solve   x+y  = 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+y  = 0 and calculate its properties

Graph of a Straight Line :

Calculate the Y-Intercept :

Notice that when x = 0 the value of y is 0/1 so this line "cuts" the y axis at y= 0.00000

  y-intercept = 0/1  =  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 -2.000. So, for a change of 2.000 in x (The change in x is sometimes referred to as "RUN") we get a change of -2.000 - -0.000 = -2.000 in y. (The change in y is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

    Slope     = -2.000/2.000 = -1.000 

Equation of a Straight Line

 2.3     Solve   x-y  = 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-y  = 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/-1 so this line "cuts" the y axis at y=-0.00000

  y-intercept = 0/-1  = -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 2.000. So, for a change of 2.000 in x (The change in x is sometimes referred to as "RUN") we get a change of 2.000 - 0.000 = 2.000 in y. (The change in y is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

     Slope     = 1

Geometric figure: Two Straight Lines

1.   Slope = 1
2.   x-intercept = 0/1 = 0.00000
3.   y-intercept = 0/-1 = -0.00000
4.   Slope = -2.000/2.000 = -1.000
5.   x-intercept = 0/1 = 0.00000
6.   y-intercept = 0/1 = 0.00000