Whakaoti i te 1/x-3-1/x-4=1/x-5-1/x-6

$$\frac { 1 } { x - 3 } - \frac { 1 } { x - 4 } = \frac { 1 } { x - 5 } - \frac { 1 } { x - 6 }$$

Whakaoti mō x

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 3,4,5,6 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-6\right)\left(x-5\right)\left(x-4\right)\left(x-3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-3,x-4,x-5,x-6.

$$\left(x-6\right)\left(x-5\right)\left(x-4\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-6 ki te x-5 ka whakakotahi i ngā kupu rite.

$$\left(x^{2}-11x+30\right)\left(x-4\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x^{2}-11x+30 ki te x-4 ka whakakotahi i ngā kupu rite.

$$x^{3}-15x^{2}+74x-120-\left(x-6\right)\left(x-5\right)\left(x-3\right)=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-6 ki te x-5 ka whakakotahi i ngā kupu rite.

$$x^{3}-15x^{2}+74x-120-\left(x^{2}-11x+30\right)\left(x-3\right)=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x^{2}-11x+30 ki te x-3 ka whakakotahi i ngā kupu rite.

$$x^{3}-15x^{2}+74x-120-\left(x^{3}-14x^{2}+63x-90\right)=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Hei kimi i te tauaro o x^{3}-14x^{2}+63x-90, kimihia te tauaro o ia taurangi.

$$x^{3}-15x^{2}+74x-120-x^{3}+14x^{2}-63x+90=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Pahekotia te x^{3} me -x^{3}, ka 0.

$$-15x^{2}+74x-120+14x^{2}-63x+90=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Pahekotia te -15x^{2} me 14x^{2}, ka -x^{2}.

$$-x^{2}+74x-120-63x+90=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Pahekotia te 74x me -63x, ka 11x.

$$-x^{2}+11x-120+90=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Tāpirihia te -120 ki te 90, ka -30.

$$-x^{2}+11x-30=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-6 ki te x-4 ka whakakotahi i ngā kupu rite.

$$-x^{2}+11x-30=\left(x^{2}-10x+24\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x^{2}-10x+24 ki te x-3 ka whakakotahi i ngā kupu rite.

$$-x^{2}+11x-30=x^{3}-13x^{2}+54x-72-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-5 ki te x-4 ka whakakotahi i ngā kupu rite.

$$-x^{2}+11x-30=x^{3}-13x^{2}+54x-72-\left(x^{2}-9x+20\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x^{2}-9x+20 ki te x-3 ka whakakotahi i ngā kupu rite.

$$-x^{2}+11x-30=x^{3}-13x^{2}+54x-72-\left(x^{3}-12x^{2}+47x-60\right)$$

Hei kimi i te tauaro o x^{3}-12x^{2}+47x-60, kimihia te tauaro o ia taurangi.

$$-x^{2}+11x-30=x^{3}-13x^{2}+54x-72-x^{3}+12x^{2}-47x+60$$

Pahekotia te x^{3} me -x^{3}, ka 0.

$$-x^{2}+11x-30=-13x^{2}+54x-72+12x^{2}-47x+60$$

Pahekotia te -13x^{2} me 12x^{2}, ka -x^{2}.

$$-x^{2}+11x-30=-x^{2}+54x-72-47x+60$$

Pahekotia te 54x me -47x, ka 7x.

$$-x^{2}+11x-30=-x^{2}+7x-72+60$$

Tāpirihia te -72 ki te 60, ka -12.

$$-x^{2}+11x-30=-x^{2}+7x-12$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$-x^{2}+11x-30+x^{2}=7x-12$$

Pahekotia te -x^{2} me x^{2}, ka 0.

$$11x-30=7x-12$$

Tangohia te 7x mai i ngā taha e rua.

$$11x-30-7x=-12$$

Pahekotia te 11x me -7x, ka 4x.

$$4x-30=-12$$

Me tāpiri te 30 ki ngā taha e rua.

$$4x=-12+30$$

Tāpirihia te -12 ki te 30, ka 18.

$$4x=18$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{18}{4}$$

Whakahekea te hautanga \frac{18}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{9}{2}$$