Question

2x-y=15 over 4x+3y=-5. What's is the solution to the system

Answer 1

Equation at the end of step  1  : 15 (2x - y) - ((—— • x) + 3y) = 0 4 Step  2  :Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a whole to a fraction 

Rewrite the whole as a fraction using  4  as the denominator :

3y 3y • 4 3y = —— = —————— 1 4

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 :

 2.2       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:

15x + 3y • 4 15x + 12y ———————————— = ————————— 4 4 Equation at the end of step  2  : (15x + 12y) (2x - y) - ——————————— = 0 4 Step  3  :Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole 

Rewrite the whole as a fraction using  4  as the denominator :

2x - y (2x - y) • 4 2x - y = —————— = ———————————— 1 4 Step  4  :Pulling out like terms :

 4.1     Pull out like factors :

   15x + 12y  =   3 • (5x + 4y) 

Adding fractions that have a common denominator :

 4.2       Adding up the two equivalent fractions 

(2x-y) • 4 - (3 • (5x+4y)) -7x - 16y —————————————————————————— = ————————— 4 4 Step  5  :Pulling out like terms :

 5.1     Pull out like factors :

   -7x - 16y  =   -1 • (7x + 16y) 

Equation at the end of step  5  : -7x - 16y ————————— = 0 4 Step  6  :When a fraction equals zero : 6.1    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:

-7x-16y ——————— • 4 = 0 • 4 4

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

The equation now takes the shape :
   -7x-16y  = 0

Equation of a Straight Line

 6.2     Solve   -7x-16y  = 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  -7x-16y  = 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/-16 so this line "cuts" the y axis at y=-0.00000

  y-intercept = 0/-16 = -0.00000 Calculate the X-Intercept :

When y = 0 the value of x is 0/-7 Our line therefore "cuts" the x axis at x=-0.00000

  x-intercept = 0/-7 = -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 -0.875. So, for a change of 2.000 in x (The change in x is sometimes referred to as "RUN") we get a change of -0.875 - 0.000 = -0.875 in y. (The change in y is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

  Slope = -0.875/2.000 = -0.438 Geometric figure: Straight Line  Slope = -0.875/2.000 = -0.438  x-intercept = 0/-7 = -0.00000  y-intercept = 0/-16 = -0.00000

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