Whakaoti i te {l}{7x+5y=60}{6x+4y=62}

$$\left\{ \begin{array} { l } { 7 x + 5 y = 60 } \\ { 6 x + 4 y = 62 } \end{array} \right.$$

Whakaoti mō x, y

$$x=35$$

$$y=-37$$

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.

$$7x+5y=60,6x+4y=62$$

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

$$7x+5y=60$$

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

$$7x=-5y+60$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-5y+60\right)$$

Whakareatia \frac{1}{7} ki te -5y+60.

$$x=-\frac{5}{7}y+\frac{60}{7}$$

Whakakapia te \frac{-5y+60}{7} mō te x ki tērā atu whārite, 6x+4y=62.

$$6\left(-\frac{5}{7}y+\frac{60}{7}\right)+4y=62$$

Whakareatia 6 ki te \frac{-5y+60}{7}.

$$-\frac{30}{7}y+\frac{360}{7}+4y=62$$

Tāpiri -\frac{30y}{7} ki te 4y.

$$-\frac{2}{7}y+\frac{360}{7}=62$$

Me tango \frac{360}{7} mai i ngā taha e rua o te whārite.

$$-\frac{2}{7}y=\frac{74}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{2}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-37$$

Whakaurua te -37 mō y ki x=-\frac{5}{7}y+\frac{60}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{7}\left(-37\right)+\frac{60}{7}$$

Whakareatia -\frac{5}{7} ki te -37.

$$x=\frac{185+60}{7}$$

Tāpiri \frac{60}{7} ki te \frac{185}{7} 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.

$$x=35$$

Kua oti te pūnaha te whakatau.

$$x=35,y=-37$$

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.

$$7x+5y=60,6x+4y=62$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&5\\6&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}60\\62\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&5\\6&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&5\\6&4\end{matrix}\right))\left(\begin{matrix}7&5\\6&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&5\\6&4\end{matrix}\right))\left(\begin{matrix}60\\62\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&5\\6&4\end{matrix}\right))\left(\begin{matrix}60\\62\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&5\\6&4\end{matrix}\right))\left(\begin{matrix}60\\62\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{7\times 4-5\times 6}&-\frac{5}{7\times 4-5\times 6}\\-\frac{6}{7\times 4-5\times 6}&\frac{7}{7\times 4-5\times 6}\end{matrix}\right)\left(\begin{matrix}60\\62\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2&\frac{5}{2}\\3&-\frac{7}{2}\end{matrix}\right)\left(\begin{matrix}60\\62\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\times 60+\frac{5}{2}\times 62\\3\times 60-\frac{7}{2}\times 62\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}35\\-37\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=35,y=-37$$

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.

$$7x+5y=60,6x+4y=62$$

Kia ōrite ai a 7x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 7.

$$6\times 7x+6\times 5y=6\times 60,7\times 6x+7\times 4y=7\times 62$$

Whakarūnātia.

$$42x+30y=360,42x+28y=434$$

Me tango 42x+28y=434 mai i 42x+30y=360 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$42x-42x+30y-28y=360-434$$

Tāpiri 42x ki te -42x. Ka whakakore atu ngā kupu 42x me -42x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$30y-28y=360-434$$

Tāpiri 30y ki te -28y.

$$2y=360-434$$

Tāpiri 360 ki te -434.

$$2y=-74$$

Whakawehea ngā taha e rua ki te 2.

$$y=-37$$

Whakaurua te -37 mō y ki 6x+4y=62. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+4\left(-37\right)=62$$

Whakareatia 4 ki te -37.

$$6x-148=62$$

Me tāpiri 148 ki ngā taha e rua o te whārite.

$$6x=210$$

Whakawehea ngā taha e rua ki te 6.

$$x=35$$

Kua oti te pūnaha te whakatau.

$$x=35,y=-37$$