Lahenda x+1/x+2+x+6/x+7=x+2/x+3+x+5/x+6

$$\frac { x + 1 } { x + 2 } + \frac { x + 6 } { x + 7 } = \frac { x + 2 } { x + 3 } + \frac { x + 5 } { x + 6 }$$

Lahendage ja leidke x

$$x = -\frac{9}{2} = -4\frac{1}{2} = -4,5$$

Juhised lineaarvõrrandi lahendamiseks

Muutuja x ei tohi võrduda ühegagi väärtustest -7,-6,-3,-2, kuna nulliga jagamist pole määratletud. Korrutage võrrandi mõlemad pooled arvuga \left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x+7\right), mis on arvu x+2,x+7,x+3,x+6 vähim ühiskordne.

$$\left(x+3\right)\left(x+6\right)\left(x+7\right)\left(x+1\right)+\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x+6\right)=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Kasutage distributiivsusomadust, et korrutada x+3 ja x+6, ning koondage sarnased liikmed.

$$\left(x^{2}+9x+18\right)\left(x+7\right)\left(x+1\right)+\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x+6\right)=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Kasutage distributiivsusomadust, et korrutada x^{2}+9x+18 ja x+7, ning koondage sarnased liikmed.

$$\left(x^{3}+16x^{2}+81x+126\right)\left(x+1\right)+\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x+6\right)=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Kasutage distributiivsusomadust, et korrutada x^{3}+16x^{2}+81x+126 ja x+1, ning koondage sarnased liikmed.

$$x^{4}+17x^{3}+97x^{2}+207x+126+\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x+6\right)=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Korrutage x+6 ja x+6, et leida \left(x+6\right)^{2}.

$$x^{4}+17x^{3}+97x^{2}+207x+126+\left(x+2\right)\left(x+3\right)\left(x+6\right)^{2}=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Kasutage kaksliikme \left(x+6\right)^{2} arendamiseks binoomvalemit \left(a+b\right)^{2}=a^{2}+2ab+b^{2}.

$$x^{4}+17x^{3}+97x^{2}+207x+126+\left(x+2\right)\left(x+3\right)\left(x^{2}+12x+36\right)=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Kasutage distributiivsusomadust, et korrutada x+2 ja x+3, ning koondage sarnased liikmed.

$$x^{4}+17x^{3}+97x^{2}+207x+126+\left(x^{2}+5x+6\right)\left(x^{2}+12x+36\right)=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Kasutage distributiivsusomadust, et korrutada x^{2}+5x+6 ja x^{2}+12x+36, ning koondage sarnased liikmed.

$$x^{4}+17x^{3}+97x^{2}+207x+126+x^{4}+17x^{3}+102x^{2}+252x+216=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Kombineerige x^{4} ja x^{4}, et leida 2x^{4}.

$$2x^{4}+17x^{3}+97x^{2}+207x+126+17x^{3}+102x^{2}+252x+216=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Kombineerige 17x^{3} ja 17x^{3}, et leida 34x^{3}.

$$2x^{4}+34x^{3}+97x^{2}+207x+126+102x^{2}+252x+216=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Kombineerige 97x^{2} ja 102x^{2}, et leida 199x^{2}.

$$2x^{4}+34x^{3}+199x^{2}+207x+126+252x+216=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Kombineerige 207x ja 252x, et leida 459x.

$$2x^{4}+34x^{3}+199x^{2}+459x+126+216=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Liitke 126 ja 216, et leida 342.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Korrutage x+2 ja x+2, et leida \left(x+2\right)^{2}.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=\left(x+2\right)^{2}\left(x+6\right)\left(x+7\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Kasutage kaksliikme \left(x+2\right)^{2} arendamiseks binoomvalemit \left(a+b\right)^{2}=a^{2}+2ab+b^{2}.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=\left(x^{2}+4x+4\right)\left(x+6\right)\left(x+7\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Kasutage distributiivsusomadust, et korrutada x^{2}+4x+4 ja x+6, ning koondage sarnased liikmed.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=\left(x^{3}+10x^{2}+28x+24\right)\left(x+7\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Kasutage distributiivsusomadust, et korrutada x^{3}+10x^{2}+28x+24 ja x+7, ning koondage sarnased liikmed.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=x^{4}+17x^{3}+98x^{2}+220x+168+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Kasutage distributiivsusomadust, et korrutada x+2 ja x+3, ning koondage sarnased liikmed.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=x^{4}+17x^{3}+98x^{2}+220x+168+\left(x^{2}+5x+6\right)\left(x+7\right)\left(x+5\right)$$

Kasutage distributiivsusomadust, et korrutada x^{2}+5x+6 ja x+7, ning koondage sarnased liikmed.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=x^{4}+17x^{3}+98x^{2}+220x+168+\left(x^{3}+12x^{2}+41x+42\right)\left(x+5\right)$$

Kasutage distributiivsusomadust, et korrutada x^{3}+12x^{2}+41x+42 ja x+5, ning koondage sarnased liikmed.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=x^{4}+17x^{3}+98x^{2}+220x+168+x^{4}+17x^{3}+101x^{2}+247x+210$$

Kombineerige x^{4} ja x^{4}, et leida 2x^{4}.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=2x^{4}+17x^{3}+98x^{2}+220x+168+17x^{3}+101x^{2}+247x+210$$

Kombineerige 17x^{3} ja 17x^{3}, et leida 34x^{3}.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=2x^{4}+34x^{3}+98x^{2}+220x+168+101x^{2}+247x+210$$

Kombineerige 98x^{2} ja 101x^{2}, et leida 199x^{2}.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=2x^{4}+34x^{3}+199x^{2}+220x+168+247x+210$$

Kombineerige 220x ja 247x, et leida 467x.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=2x^{4}+34x^{3}+199x^{2}+467x+168+210$$

Liitke 168 ja 210, et leida 378.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=2x^{4}+34x^{3}+199x^{2}+467x+378$$

Lahutage mõlemast poolest 2x^{4}.

$$2x^{4}+34x^{3}+199x^{2}+459x+342-2x^{4}=34x^{3}+199x^{2}+467x+378$$

Kombineerige 2x^{4} ja -2x^{4}, et leida 0.

$$34x^{3}+199x^{2}+459x+342=34x^{3}+199x^{2}+467x+378$$

Lahutage mõlemast poolest 34x^{3}.

$$34x^{3}+199x^{2}+459x+342-34x^{3}=199x^{2}+467x+378$$

Kombineerige 34x^{3} ja -34x^{3}, et leida 0.

$$199x^{2}+459x+342=199x^{2}+467x+378$$

Lahutage mõlemast poolest 199x^{2}.

$$199x^{2}+459x+342-199x^{2}=467x+378$$

Kombineerige 199x^{2} ja -199x^{2}, et leida 0.

$$459x+342=467x+378$$

Lahutage mõlemast poolest 467x.

$$459x+342-467x=378$$

Kombineerige 459x ja -467x, et leida -8x.

$$-8x+342=378$$

Lahutage mõlemast poolest 342.

$$-8x=378-342$$

Lahutage 342 väärtusest 378, et leida 36.

$$-8x=36$$

Jagage mõlemad pooled -8-ga.

$$x=\frac{36}{-8}$$

Taandage murd \frac{36}{-8} vähimale ühiskordsele, eraldades ja taandades arvu 4.

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