Question

Korzystając z własności proporcji, rozwiąż równanie:

a

b

c

Answer 1

a)

1/3x = (2x-3)/2x                                                Z: x ≠ 0

                                                                        D = R \ {0}

3x(2x-3) = 2x

6x²-9x = 2x

6x²-9x-2x = 0

6x²-11x = 0

x(6x-11) = 0

x = 0  ∉ D

6x-11 = 0

6x = 11  /:6

x = 11/6

x = 1⅚

Odp. x = 1⅚

 

b)

(4x-3)/(2x-3) = (5x-3)/(x+1)

                            Z: x+1 ≠ 0

                                x ≠ -1

                               2x-3 ≠ 0

                                 x ≠ 1½

                             D = R \ {-1;1½}

 

(2x-3)(5x-3) = (4x-3)(x+1)

10x²-6x-15x+9 = 4x²+4x-3x-3

10x²-21x+9 = 4x²+x-3

10x²-4x²-21x-x+9+3 = 0

6x²-22x+12 = 0  /:2

3x²-11x+6 = 0

Δ = 121-72 = 49

√Δ = √49 = 7 

x1 = (11-7)/6 = 2/3

x2 = (11+7)/6 = 3

Odp. x = 2/3   v   x = 3

 

c)

(x-6)/(x-5) = (2x-11)/(x-1)

                            Z: x-1 ≠ 0

                                x ≠ 1

                                x-5 ≠ 0

                                x ≠ 5

                            D = R \ {1;5}

(x-5)(2x-11) = (x-1)(x-6)

2x²-11x-10x+55 = x²-6x-x+6

2x²-21x+55 = x²-7x+6  

2x²-x²-21x+7x+55-6 = 0

x²-14x+49 = 0

Δ = 196-196 = 0

x = 14/2 = 7

Odp. x = 7