Løs 1/2t+1+1/27-i=4/z+2

$$\frac { 1 } { 2 t + 1 } + \frac { 1 } { 27 - i } = \frac { 4 } { z + 2 }$$

Løs for t

$$t=-\frac{\left(757+i\right)z+\left(-1406+2i\right)}{2\left(\left(27+i\right)z+\left(-2866+2i\right)\right)}$$

$$z\neq -2\text{ and }z\neq 106-4i$$

Trinn for å løse lineære ligninger

Variabelen t kan ikke være lik -\frac{1}{2} siden divisjon med null ikke er definert. Multipliser begge sider av formelen med \left(z+2\right)\left(2t+1\right), som er den minste fellesnevneren av 2t+1,z+2.

$$z+2+\left(z+2\right)\left(2t+1\right)\times \frac{1}{27-i}=\left(2t+1\right)\times 4$$

Multipliserer både teller og nevner av \frac{1}{27-i} med komplekskonjugatet av nevneren 27+i.

$$z+2+\left(z+2\right)\left(2t+1\right)\times \frac{1\left(27+i\right)}{\left(27-i\right)\left(27+i\right)}=\left(2t+1\right)\times 4$$

Utfør multiplikasjonene i \frac{1\left(27+i\right)}{\left(27-i\right)\left(27+i\right)}.

$$z+2+\left(z+2\right)\left(2t+1\right)\times \frac{27+i}{730}=\left(2t+1\right)\times 4$$

Del 27+i på 730 for å få \frac{27}{730}+\frac{1}{730}i.

$$z+2+\left(z+2\right)\left(2t+1\right)\left(\frac{27}{730}+\frac{1}{730}i\right)=\left(2t+1\right)\times 4$$

Bruk den distributive lov til å multiplisere z+2 med 2t+1.

$$z+2+\left(2zt+z+4t+2\right)\left(\frac{27}{730}+\frac{1}{730}i\right)=\left(2t+1\right)\times 4$$

Bruk den distributive lov til å multiplisere 2zt+z+4t+2 med \frac{27}{730}+\frac{1}{730}i.

$$z+2+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{27}{730}+\frac{1}{730}i\right)z+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\left(\frac{27}{365}+\frac{1}{365}i\right)=\left(2t+1\right)\times 4$$

Utfør addisjonene i 2+\left(\frac{27}{365}+\frac{1}{365}i\right).

$$z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{27}{730}+\frac{1}{730}i\right)z+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=\left(2t+1\right)\times 4$$

Kombiner z og \left(\frac{27}{730}+\frac{1}{730}i\right)z for å få \left(\frac{757}{730}+\frac{1}{730}i\right)z.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=\left(2t+1\right)\times 4$$

Bruk den distributive lov til å multiplisere 2t+1 med 4.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=8t+4$$

Trekk fra 8t fra begge sider.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i-8t=4$$

Kombiner \left(\frac{54}{365}+\frac{2}{365}i\right)t og -8t for å få \left(-\frac{2866}{365}+\frac{2}{365}i\right)t.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(-\frac{2866}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=4$$

Trekk fra \left(\frac{757}{730}+\frac{1}{730}i\right)z fra begge sider.

$$\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(-\frac{2866}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=4-\left(\frac{757}{730}+\frac{1}{730}i\right)z$$

Trekk fra \frac{757}{365} fra begge sider.

$$\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(-\frac{2866}{365}+\frac{2}{365}i\right)t+\frac{1}{365}i=4-\left(\frac{757}{730}+\frac{1}{730}i\right)z-\frac{757}{365}$$

Trekk fra \frac{1}{365}i fra begge sider.

$$\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(-\frac{2866}{365}+\frac{2}{365}i\right)t=4-\left(\frac{757}{730}+\frac{1}{730}i\right)z-\frac{757}{365}-\frac{1}{365}i$$

Utfør addisjonene i 4-\frac{757}{365}-\frac{1}{365}i.

$$\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(-\frac{2866}{365}+\frac{2}{365}i\right)t=-\left(\frac{757}{730}+\frac{1}{730}i\right)z+\frac{703}{365}-\frac{1}{365}i$$

Multipliser -1 med \frac{757}{730}+\frac{1}{730}i for å få -\frac{757}{730}-\frac{1}{730}i.

$$\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(-\frac{2866}{365}+\frac{2}{365}i\right)t=\left(-\frac{757}{730}-\frac{1}{730}i\right)z+\frac{703}{365}-\frac{1}{365}i$$

Kombiner alle ledd som inneholder t.

$$\left(\left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right)\right)t=\left(-\frac{757}{730}-\frac{1}{730}i\right)z+\frac{703}{365}-\frac{1}{365}i$$

Ligningen er i standardform.

$$\left(\left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right)\right)t=\left(-\frac{757}{730}-\frac{1}{730}i\right)z+\left(\frac{703}{365}-\frac{1}{365}i\right)$$

Del begge sidene på \left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right).

$$\frac{\left(\left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right)\right)t}{\left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right)}=\frac{\left(-\frac{757}{730}-\frac{1}{730}i\right)z+\left(\frac{703}{365}-\frac{1}{365}i\right)}{\left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right)}$$

Hvis du deler på \left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right), gjør du om gangingen med \left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right).

$$t=\frac{\left(-\frac{757}{730}-\frac{1}{730}i\right)z+\left(\frac{703}{365}-\frac{1}{365}i\right)}{\left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right)}$$

Del \left(-\frac{757}{730}-\frac{1}{730}i\right)z+\left(\frac{703}{365}-\frac{1}{365}i\right) på \left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right).

$$t=\frac{\left(-757-i\right)z+\left(1406-2i\right)}{2\left(\left(27+i\right)z+\left(-2866+2i\right)\right)}$$

Variabelen t kan ikke være lik -\frac{1}{2}.

$$t=\frac{\left(-757-i\right)z+\left(1406-2i\right)}{2\left(\left(27+i\right)z+\left(-2866+2i\right)\right)}\text{, }t\neq -\frac{1}{2}$$

Løs for z

$$z=-\frac{2\left(\left(2866-2i\right)t+\left(703-i\right)\right)}{\left(-54-2i\right)t+\left(-757-i\right)}$$

$$t\neq -\frac{1}{2}\text{ and }t\neq -14+\frac{1}{2}i$$

Trinn for å løse lineære ligninger

Variabelen z kan ikke være lik -2 siden divisjon med null ikke er definert. Multipliser begge sider av formelen med \left(z+2\right)\left(2t+1\right), som er den minste fellesnevneren av 2t+1,z+2.

$$z+2+\left(z+2\right)\left(2t+1\right)\times \frac{1}{27-i}=\left(2t+1\right)\times 4$$

Multipliserer både teller og nevner av \frac{1}{27-i} med komplekskonjugatet av nevneren 27+i.

$$z+2+\left(z+2\right)\left(2t+1\right)\times \frac{1\left(27+i\right)}{\left(27-i\right)\left(27+i\right)}=\left(2t+1\right)\times 4$$

Utfør multiplikasjonene i \frac{1\left(27+i\right)}{\left(27-i\right)\left(27+i\right)}.

$$z+2+\left(z+2\right)\left(2t+1\right)\times \frac{27+i}{730}=\left(2t+1\right)\times 4$$

Del 27+i på 730 for å få \frac{27}{730}+\frac{1}{730}i.

$$z+2+\left(z+2\right)\left(2t+1\right)\left(\frac{27}{730}+\frac{1}{730}i\right)=\left(2t+1\right)\times 4$$

Bruk den distributive lov til å multiplisere z+2 med 2t+1.

$$z+2+\left(2zt+z+4t+2\right)\left(\frac{27}{730}+\frac{1}{730}i\right)=\left(2t+1\right)\times 4$$

Bruk den distributive lov til å multiplisere 2zt+z+4t+2 med \frac{27}{730}+\frac{1}{730}i.

$$z+2+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{27}{730}+\frac{1}{730}i\right)z+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\left(\frac{27}{365}+\frac{1}{365}i\right)=\left(2t+1\right)\times 4$$

Utfør addisjonene i 2+\left(\frac{27}{365}+\frac{1}{365}i\right).

$$z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{27}{730}+\frac{1}{730}i\right)z+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=\left(2t+1\right)\times 4$$

Kombiner z og \left(\frac{27}{730}+\frac{1}{730}i\right)z for å få \left(\frac{757}{730}+\frac{1}{730}i\right)z.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=\left(2t+1\right)\times 4$$

Bruk den distributive lov til å multiplisere 2t+1 med 4.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=8t+4$$

Trekk fra \left(\frac{54}{365}+\frac{2}{365}i\right)t fra begge sider.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\frac{757}{365}+\frac{1}{365}i=8t+4-\left(\frac{54}{365}+\frac{2}{365}i\right)t$$

Kombiner 8t og \left(-\frac{54}{365}-\frac{2}{365}i\right)t for å få \left(\frac{2866}{365}-\frac{2}{365}i\right)t.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\frac{757}{365}+\frac{1}{365}i=\left(\frac{2866}{365}-\frac{2}{365}i\right)t+4$$

Trekk fra \frac{757}{365} fra begge sider.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\frac{1}{365}i=\left(\frac{2866}{365}-\frac{2}{365}i\right)t+4-\frac{757}{365}$$

Trekk fra \frac{757}{365} fra 4 for å få \frac{703}{365}.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\frac{1}{365}i=\left(\frac{2866}{365}-\frac{2}{365}i\right)t+\frac{703}{365}$$

Trekk fra \frac{1}{365}i fra begge sider.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt=\left(\frac{2866}{365}-\frac{2}{365}i\right)t+\frac{703}{365}-\frac{1}{365}i$$

Kombiner alle ledd som inneholder z.

$$\left(\frac{757}{730}+\frac{1}{730}i+\left(\frac{27}{365}+\frac{1}{365}i\right)t\right)z=\left(\frac{2866}{365}-\frac{2}{365}i\right)t+\frac{703}{365}-\frac{1}{365}i$$

Ligningen er i standardform.

$$\left(\left(\frac{27}{365}+\frac{1}{365}i\right)t+\left(\frac{757}{730}+\frac{1}{730}i\right)\right)z=\left(\frac{2866}{365}-\frac{2}{365}i\right)t+\left(\frac{703}{365}-\frac{1}{365}i\right)$$

Del begge sidene på \frac{757}{730}+\frac{1}{730}i+\left(\frac{27}{365}+\frac{1}{365}i\right)t.

$$\frac{\left(\left(\frac{27}{365}+\frac{1}{365}i\right)t+\left(\frac{757}{730}+\frac{1}{730}i\right)\right)z}{\left(\frac{27}{365}+\frac{1}{365}i\right)t+\left(\frac{757}{730}+\frac{1}{730}i\right)}=\frac{\left(\frac{2866}{365}-\frac{2}{365}i\right)t+\left(\frac{703}{365}-\frac{1}{365}i\right)}{\left(\frac{27}{365}+\frac{1}{365}i\right)t+\left(\frac{757}{730}+\frac{1}{730}i\right)}$$

Hvis du deler på \frac{757}{730}+\frac{1}{730}i+\left(\frac{27}{365}+\frac{1}{365}i\right)t, gjør du om gangingen med \frac{757}{730}+\frac{1}{730}i+\left(\frac{27}{365}+\frac{1}{365}i\right)t.

$$z=\frac{\left(\frac{2866}{365}-\frac{2}{365}i\right)t+\left(\frac{703}{365}-\frac{1}{365}i\right)}{\left(\frac{27}{365}+\frac{1}{365}i\right)t+\left(\frac{757}{730}+\frac{1}{730}i\right)}$$

Del \left(\frac{2866}{365}-\frac{2}{365}i\right)t+\left(\frac{703}{365}-\frac{1}{365}i\right) på \frac{757}{730}+\frac{1}{730}i+\left(\frac{27}{365}+\frac{1}{365}i\right)t.

$$z=\frac{2\left(\left(2866-2i\right)t+\left(703-i\right)\right)}{\left(54+2i\right)t+\left(757+i\right)}$$

Variabelen z kan ikke være lik -2.

$$z=\frac{2\left(\left(2866-2i\right)t+\left(703-i\right)\right)}{\left(54+2i\right)t+\left(757+i\right)}\text{, }z\neq -2$$