Mister Exam

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(2y)/(y-3)=(3y+3)/y equation

The teacher will be very surprised to see your correct solution 😉

You have entered [src]

 2*y    3*y + 3
----- = -------
y - 3      y

\frac{2 y}{y - 3} = \frac{3 y + 3}{y}

Simplify

Detail solution

Given the equation:

\frac{2 y}{y - 3} = \frac{3 y + 3}{y}

Multiply the equation sides by the denominators:
-3 + y and y
we get:

\frac{2 y \left(y - 3\right)}{y - 3} = \frac{\left(y - 3\right) \left(3 y + 3\right)}{y}

2 y = 3 y - 6 - \frac{9}{y}

y 2 y = y \left(3 y - 6 - \frac{9}{y}\right)

2 y^{2} = 3 y^{2} - 6 y - 9

Move right part of the equation to
left part with negative sign.

The equation is transformed from

2 y^{2} = 3 y^{2} - 6 y - 9

- y^{2} + 6 y + 9 = 0

This equation is of the form

a*y^2 + b*y + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:

y_{1} = \frac{\sqrt{D} - b}{2 a}

y_{2} = \frac{- \sqrt{D} - b}{2 a}

where D = b^2 - 4*a*c - it is the discriminant.
Because

a = -1

b = 6

c = 9

, then

D = b^2 - 4 * a * c =

(6)^2 - 4 * (-1) * (9) = 72

Because D > 0, then the equation has two roots.

y1 = (-b + sqrt(D)) / (2*a)

y2 = (-b - sqrt(D)) / (2*a)

y_{1} = 3 - 3 \sqrt{2}

y_{2} = 3 + 3 \sqrt{2}

The graph

Plot the graph f1(x)

Plot the graph f2(x)

Sum and product of roots [src]

sum

        ___           ___
3 - 3*\/ 2  + 3 + 3*\/ 2

\left(3 - 3 \sqrt{2}\right) + \left(3 + 3 \sqrt{2}\right)

6

product

/        ___\ /        ___\
\3 - 3*\/ 2 /*\3 + 3*\/ 2 /

\left(3 - 3 \sqrt{2}\right) \left(3 + 3 \sqrt{2}\right)

-9

-9

-9

Rapid solution [src]

             ___
y1 = 3 - 3*\/ 2

y_{1} = 3 - 3 \sqrt{2}

             ___
y2 = 3 + 3*\/ 2

y_{2} = 3 + 3 \sqrt{2}

y2 = 3 + 3*sqrt(2)

Numerical answer [src]

y1 = -1.24264068711929

y2 = 7.24264068711928

y2 = 7.24264068711928