Whakaoti i te {l}{4x+5y=1}{5x-7y=1}

$$\left. \begin{array} { l } { 4 x + 5 y = 1 } \\ { 5 x - 7 y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{12}{53}\approx 0.226415094$$

$$y=\frac{1}{53}\approx 0.018867925$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$4x+5y=1,5x-7y=1$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$4x+5y=1$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$4x=-5y+1$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-5y+1\right)$$

Whakareatia \frac{1}{4} ki te -5y+1.

$$x=-\frac{5}{4}y+\frac{1}{4}$$

Whakakapia te \frac{-5y+1}{4} mō te x ki tērā atu whārite, 5x-7y=1.

$$5\left(-\frac{5}{4}y+\frac{1}{4}\right)-7y=1$$

Whakareatia 5 ki te \frac{-5y+1}{4}.

$$-\frac{25}{4}y+\frac{5}{4}-7y=1$$

Tāpiri -\frac{25y}{4} ki te -7y.

$$-\frac{53}{4}y+\frac{5}{4}=1$$

Me tango \frac{5}{4} mai i ngā taha e rua o te whārite.

$$-\frac{53}{4}y=-\frac{1}{4}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{53}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{1}{53}$$

Whakaurua te \frac{1}{53} mō y ki x=-\frac{5}{4}y+\frac{1}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{4}\times \frac{1}{53}+\frac{1}{4}$$

Whakareatia -\frac{5}{4} ki te \frac{1}{53} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{5}{212}+\frac{1}{4}$$

Tāpiri \frac{1}{4} ki te -\frac{5}{212} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{12}{53}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{12}{53},y=\frac{1}{53}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$4x+5y=1,5x-7y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&5\\5&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&5\\5&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&5\\5&-7\end{matrix}\right))\left(\begin{matrix}4&5\\5&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\5&-7\end{matrix}\right))\left(\begin{matrix}1\\1\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\5&-7\end{matrix}\right))\left(\begin{matrix}1\\1\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\5&-7\end{matrix}\right))\left(\begin{matrix}1\\1\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{4\left(-7\right)-5\times 5}&-\frac{5}{4\left(-7\right)-5\times 5}\\-\frac{5}{4\left(-7\right)-5\times 5}&\frac{4}{4\left(-7\right)-5\times 5}\end{matrix}\right)\left(\begin{matrix}1\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{53}&\frac{5}{53}\\\frac{5}{53}&-\frac{4}{53}\end{matrix}\right)\left(\begin{matrix}1\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7+5}{53}\\\frac{5-4}{53}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{53}\\\frac{1}{53}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{12}{53},y=\frac{1}{53}$$

Ngā Upane e Whakamahi i te Tangohanga

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$4x+5y=1,5x-7y=1$$

Kia ōrite ai a 4x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$5\times 4x+5\times 5y=5,4\times 5x+4\left(-7\right)y=4$$

Whakarūnātia.

$$20x+25y=5,20x-28y=4$$

Me tango 20x-28y=4 mai i 20x+25y=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20x-20x+25y+28y=5-4$$

Tāpiri 20x ki te -20x. Ka whakakore atu ngā kupu 20x me -20x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$25y+28y=5-4$$

Tāpiri 25y ki te 28y.

$$53y=5-4$$

Tāpiri 5 ki te -4.

$$53y=1$$

Whakawehea ngā taha e rua ki te 53.

$$y=\frac{1}{53}$$

Whakaurua te \frac{1}{53} mō y ki 5x-7y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-7\times \frac{1}{53}=1$$

Whakareatia -7 ki te \frac{1}{53}.

$$5x-\frac{7}{53}=1$$

Me tāpiri \frac{7}{53} ki ngā taha e rua o te whārite.

$$5x=\frac{60}{53}$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{12}{53}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{12}{53},y=\frac{1}{53}$$